diff --git a/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf b/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..bc3c54abd2f2d2333578033744a30ef5fd9a2400 --- /dev/null +++ b/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a9458f47cc0f45f1baffefa068d5921c0cf95128c5e6ddad858cef7718d9fab3 +size 1692730 diff --git a/.gitattributes b/.gitattributes index c2784366e145929564e510047022644468f0a55c..68485dcf893d5865f25bb7ebf2084006144c8090 100644 --- a/.gitattributes +++ b/.gitattributes @@ -2861,3 +2861,71 @@ adE1T4oBgHgl3EQfdAQO/content/2301.03189v1.pdf filter=lfs diff=lfs merge=lfs -tex KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf filter=lfs diff=lfs merge=lfs -text 29FKT4oBgHgl3EQf8C4w/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text zdE2T4oBgHgl3EQf4Qh4/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +KtE2T4oBgHgl3EQfpwgn/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +KtE2T4oBgHgl3EQfpwgn/content/2301.04031v1.pdf filter=lfs diff=lfs merge=lfs -text +L9E3T4oBgHgl3EQfBAkx/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +I9E4T4oBgHgl3EQfhQ2p/content/2301.05124v1.pdf filter=lfs diff=lfs merge=lfs -text +WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf filter=lfs diff=lfs merge=lfs -text +1dFPT4oBgHgl3EQfUjQc/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +99FAT4oBgHgl3EQfqB0k/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +w9AzT4oBgHgl3EQfCfoX/content/2301.00958v1.pdf filter=lfs diff=lfs merge=lfs -text +TNFLT4oBgHgl3EQfQC8d/content/2301.12030v1.pdf filter=lfs diff=lfs merge=lfs -text +UdE_T4oBgHgl3EQfxBzD/content/2301.08310v1.pdf filter=lfs diff=lfs merge=lfs -text +59AyT4oBgHgl3EQfQfYL/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +KdFRT4oBgHgl3EQf0zhG/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +s9E4T4oBgHgl3EQfVwyP/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +0dE2T4oBgHgl3EQf4wg0/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +idE5T4oBgHgl3EQfFw57/content/2301.05425v1.pdf filter=lfs diff=lfs merge=lfs -text +HtE2T4oBgHgl3EQf_AnW/content/2301.04245v1.pdf filter=lfs diff=lfs merge=lfs -text +d9AzT4oBgHgl3EQfaPwm/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +WtFOT4oBgHgl3EQf7zTC/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +ZNFJT4oBgHgl3EQf7S0N/content/2301.11677v1.pdf filter=lfs diff=lfs merge=lfs -text +59AzT4oBgHgl3EQfvP0x/content/2301.01702v1.pdf filter=lfs diff=lfs merge=lfs -text +99E1T4oBgHgl3EQfCgIZ/content/2301.02864v1.pdf filter=lfs diff=lfs merge=lfs -text +39E4T4oBgHgl3EQfAwsS/content/2301.04845v1.pdf filter=lfs diff=lfs merge=lfs -text +MtFJT4oBgHgl3EQfzS0r/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +x9AzT4oBgHgl3EQfCfrw/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +ctFJT4oBgHgl3EQf-S0l/content/2301.11691v1.pdf filter=lfs diff=lfs merge=lfs -text +H9AyT4oBgHgl3EQfrvmy/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +LNAyT4oBgHgl3EQfgPhD/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +adE1T4oBgHgl3EQfdAQO/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +o9E0T4oBgHgl3EQfaAAE/content/2301.02327v1.pdf filter=lfs diff=lfs merge=lfs -text +69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf filter=lfs diff=lfs merge=lfs -text +3NFIT4oBgHgl3EQf5iun/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +LtFOT4oBgHgl3EQf0TSa/content/2301.12935v1.pdf filter=lfs diff=lfs merge=lfs -text +f9E4T4oBgHgl3EQfRgxA/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +ONE0T4oBgHgl3EQfjgHF/content/2301.02461v1.pdf filter=lfs diff=lfs merge=lfs -text +M9E1T4oBgHgl3EQftQXp/content/2301.03376v1.pdf filter=lfs diff=lfs merge=lfs -text +BtE4T4oBgHgl3EQfeA0g/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +ZNFJT4oBgHgl3EQf7S0N/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +S9E2T4oBgHgl3EQfWwd9/content/2301.03837v1.pdf filter=lfs diff=lfs merge=lfs -text +a9AyT4oBgHgl3EQfW_fi/content/2301.00176v1.pdf filter=lfs diff=lfs merge=lfs -text +39E4T4oBgHgl3EQfAwsS/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +59AzT4oBgHgl3EQfvP0x/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +idE5T4oBgHgl3EQfFw57/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +69E0T4oBgHgl3EQfwAF2/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +19E4T4oBgHgl3EQfzg22/content/2301.05275v1.pdf filter=lfs diff=lfs merge=lfs -text +ONE0T4oBgHgl3EQfjgHF/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +ctFJT4oBgHgl3EQf-S0l/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +jNAzT4oBgHgl3EQfNPst/content/2301.01144v1.pdf filter=lfs diff=lfs merge=lfs -text +1NFQT4oBgHgl3EQf1DYE/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +DNE0T4oBgHgl3EQfQQC5/content/2301.02191v1.pdf filter=lfs diff=lfs merge=lfs -text +NtAyT4oBgHgl3EQfgviP/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +tNA0T4oBgHgl3EQfLf8y/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +3dAzT4oBgHgl3EQf9P73/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +BNAzT4oBgHgl3EQfhv2_/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +o9E0T4oBgHgl3EQfaAAE/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +dNFQT4oBgHgl3EQfjDY0/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +U9AzT4oBgHgl3EQfJvv5/content/2301.01087v1.pdf filter=lfs diff=lfs merge=lfs -text +ptFST4oBgHgl3EQfNzhl/content/2301.13749v1.pdf filter=lfs diff=lfs merge=lfs -text +7dAzT4oBgHgl3EQfgPzo/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +DNE0T4oBgHgl3EQfQQC5/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf filter=lfs diff=lfs merge=lfs -text +ltE1T4oBgHgl3EQfgwTU/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +dNFQT4oBgHgl3EQfjDY0/content/2301.13352v1.pdf filter=lfs diff=lfs merge=lfs -text +SdE0T4oBgHgl3EQfUgDE/content/2301.02252v1.pdf filter=lfs diff=lfs merge=lfs -text +M9E1T4oBgHgl3EQftQXp/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +CdFQT4oBgHgl3EQf_DcV/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +a9AyT4oBgHgl3EQfW_fi/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +4NFKT4oBgHgl3EQfRi0P/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +y9E4T4oBgHgl3EQfyA3N/content/2301.05263v1.pdf filter=lfs diff=lfs merge=lfs -text diff --git a/0dE2T4oBgHgl3EQf4wg0/vector_store/index.faiss b/0dE2T4oBgHgl3EQf4wg0/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..6d90d664981226303cb32eb5bef445e90a57719f --- /dev/null +++ b/0dE2T4oBgHgl3EQf4wg0/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:99fa4a8f054aa5da9a9c7d83d405c7daf3c4f5acf5cd16911ecdaa238cca363a +size 1900589 diff --git a/19E4T4oBgHgl3EQfzg22/content/2301.05275v1.pdf b/19E4T4oBgHgl3EQfzg22/content/2301.05275v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..3452887bd03279f86056a1df19b6ab1d9e5fed61 --- /dev/null +++ b/19E4T4oBgHgl3EQfzg22/content/2301.05275v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:90554dfff8a3a7d2d5016fcadcba389d9fbe2d594a0a8d4d8fb5a4715db002d5 +size 454882 diff --git a/19E4T4oBgHgl3EQfzg22/vector_store/index.pkl b/19E4T4oBgHgl3EQfzg22/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..0ec70ccac625896c8104bac72fd8e22c9b984797 --- /dev/null +++ b/19E4T4oBgHgl3EQfzg22/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b0d997d1020eddeaba30af71acde07a65958188f062f68a6852ce354a3d427e0 +size 168958 diff --git a/1NFQT4oBgHgl3EQf1DYE/vector_store/index.faiss b/1NFQT4oBgHgl3EQf1DYE/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..db0a0a810733d551d2652a6f4d8142a8b06a8112 --- /dev/null +++ b/1NFQT4oBgHgl3EQf1DYE/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c8ea2b0eba17f2fc1fac79159419e2e53a0269fa20589135bdf80166ef756b3b +size 5439533 diff --git a/1NFQT4oBgHgl3EQf1DYE/vector_store/index.pkl b/1NFQT4oBgHgl3EQf1DYE/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..7e73838f235121d6ba8f1849fa57725d0e685c9d --- /dev/null +++ b/1NFQT4oBgHgl3EQf1DYE/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0599e3b1f1924e767c23f7607d9a7aa854b64203c9e9ae5c508cf4d85a07b3fd +size 191723 diff --git a/1dFPT4oBgHgl3EQfUjQc/vector_store/index.faiss b/1dFPT4oBgHgl3EQfUjQc/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..789dae17a138e933ee38c5949b7078a3bf463c98 --- /dev/null +++ b/1dFPT4oBgHgl3EQfUjQc/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e193bcef0cecd8c99fa4fa355ec9caf1f6414b79e517d399dfe8e8aa5726546c +size 4194349 diff --git a/1tE0T4oBgHgl3EQfuQHv/content/tmp_files/2301.02604v1.pdf.txt b/1tE0T4oBgHgl3EQfuQHv/content/tmp_files/2301.02604v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b71ac2b44d92ae0c54a542b27ae62c4d6e2face6 --- /dev/null +++ b/1tE0T4oBgHgl3EQfuQHv/content/tmp_files/2301.02604v1.pdf.txt @@ -0,0 +1,1585 @@ +MNRAS 000, 1–12 (2022) +Preprint 9 January 2023 +Compiled using MNRAS LATEX style file v3.0 +A study of convective core overshooting as a function of stellar mass based +on two-dimensional hydrodynamical simulations +I. Baraffe,1,2 ★ J. Clarke,1 A. Morison,1 D. G. Vlaykov,1 T. Constantino,1 T. Goffrey,3 T. Guillet,1 +A. Le Saux1,2 and J. Pratt4 +1University of Exeter, Physics and Astronomy, EX4 4QL Exeter, UK +2École Normale Supérieure, Lyon, CRAL (UMR CNRS 5574), Université de Lyon, France +3Centre for Fusion, Space and Astrophysics, Department of Physics, University of Warwick, Coventry, CV4 7AL, UK +4Lawrence Livermore National Laboratory, 7000 East Ave, Livermore, CA 94550, USA +Accepted XXX. Received YYY +ABSTRACT +We perform two-dimensional numerical simulations of core convection for zero-age-main-sequence stars covering a mass range +from 3 𝑀⊙ to 20 𝑀⊙. The simulations are performed with the fully compressible time-implicit code MUSIC. We study the +efficiency of overshooting, which describes the ballistic process of convective flows crossing a convective boundary, as a function +of stellar mass and luminosity. We also study the impact of artificially increasing the stellar luminosity for 3 𝑀⊙ models. The +simulations cover hundreds to thousands of convective turnover timescales. Applying the framework of extreme plume events +previously developed for convective envelopes, we derive overshooting lengths as a function of stellar masses. We find that the +overshooting distance (𝑑ov) scales with the stellar luminosity (𝐿) and the convective core radius (𝑟conv). We derive a scaling law +𝑑ov ∝ 𝐿1/3𝑟1/2 +conv which is implemented in a 1D stellar evolution code and the resulting stellar models are compared to observations. +The scaling predicts values for the overshooting distance that significantly increase with stellar mass, in qualitative agreement +with observations. Quantitatively, however, the predicted values are underestimated for masses +>∼ 10𝑀⊙. Our 2D simulations +show the formation of a nearly-adiabatic layer just above the Schwarzschild boundary of the convective core, as exhibited in +recent 3D simulations of convection. The most luminous models show a growth in size with time of the nearly-adiabatic layer. +This growth seems to slow down as the upper edge of the nearly-adiabatic layer gets closer to the maximum overshooting length +and as the simulation time exceeds the typical thermal diffusive timescale in the overshooting layer. +Key words: Convection – Hydrodynamics – Stars: evolution +1 INTRODUCTION +One of the major uncertainties in stellar evolution models is the treat- +ment of mixing taking place at convective boundaries (see Stancliffe +et al. 2016). Convective motions do not abruptly stop at the classical +Schwarzschild boundary, but extend beyond it and lead to the pro- +cess of convective boundary mixing (CBM). The complex dynamics +resulting from convective flows penetrating in stable layers drives +the transport of chemical species and heat, strongly affecting the +structure and the evolution of stars. The same complex dynamics can +also drive transport of angular momentum, impacting the rotational +evolution of stars, the generation of magnetic field in their interior +and their magnetic activity. CBM affects the evolution of all stars that +develop a convective envelope, core or shell. Its treatment is one of +the oldest unsolved problems of stellar structure and evolution theory +(Shaviv & Salpeter 1973). This extra mixing could significantly alter +the size of a convective core, the lifetime of major burning phases +or the surface chemistry over a wide range of stellar masses. It can +impact the entire evolution of massive stars (𝑀 >∼ 8𝑀⊙), determin- +ing their structure before core-collapse supernova explosion and thus +★ E-mail: i.baraffe@ex.ac.uk +affecting nucleosynthetic yields which are crucial for galactic evolu- +tion studies (Arnett & Meakin 2011). There is ample observational +evidence pointing towards the need for extra internal mixing to ex- +plain a wide range of observations, such as eclipsing binaries (Claret +& Torres 2016), color-magnitude diagrams (Rosenfield et al. 2017) +or asteroseismology (Bossini et al. 2015). Rosenfield et al. (2017) +illustrate the uncertainty due to the treatment of core overshooting on +ages and on morphological changes in stellar evolution tracks, signif- +icantly impacting stellar population studies. An increasing number of +observational studies also suggests an increase of convective bound- +ary mixing efficiency with stellar mass, using eclipsing binaries (see +Claret & Torres 2019, and references therein) or Hertzsprung-Russell +diagrams of massive stars (Castro et al. 2014). In a recent study, John- +ston (2021) confirms that current stellar models with no or with little +convective boundary mixing usually under-predict the mass of con- +vective cores. While such comparisons between stellar models and +observations cannot identify a mechanism responsible for mixing at +the convective boundaries, Johnston (2021) concludes that a range of +efficiencies for the mixing mechanism(s) should be used. In addition +to CBM, additional mixing could be due to rotation (Zahn 1992) or +internal gravity waves (Schatzman 1993). The latter are connected to +CBM as they are excited at convective boundaries by turbulent con- +© 2022 The Authors +arXiv:2301.02604v1 [astro-ph.SR] 6 Jan 2023 + +2 +I. Baraffe et al. +vective motions (Press 1981; Goldreich & Kumar 1990; Lecoanet +& Quataert 2013) and penetrating flows (Rieutord & Zahn 1995; +Montalbán & Schatzman 2000; Pinçon et al. 2016). +CBM is a generic term that encompasses different processes, +namely penetration, overshooting or entrainment. The first term de- +scribes motions that cross a convective boundary and alter the back- +ground in such a way that the location of the convective boundary, +defined by the Schwarzschild or the Ledoux criterion, moves inward +or outward, resulting in the extension of the convective region. Over- +shooting usually describes convective penetrative motions that do not +alter the background but can still result in more or less efficient mixing +(Zahn 1991). In the literature, the terms overshooting and penetration +are often used interchangeably. These processes have been described +in stellar evolution models by an overshooting distance 𝑑ov and/or a +diffusion coefficient which remains constant or exponentially decays +over the overshooting length (Freytag et al. 1996). These parameters +are usually calibrated to fit observations. The temperature gradient in +the overshooting region is either set to the radiative or to the adiabatic +temperature gradient (see for example Michielsen et al. 2019). The +third term entrainment is used to characterise shear-induced turbulent +motions at the interface between the convectively stable and unstable +regions driven by convective penetrative motions (plumes or eddies). +Interfacial instabilities contribute to mixing fluids of different com- +positions and/or densities, eroding the convective boundary. This one +can then grow in time following an entrainment rate characterised +by the bulk Richardson number (Fernando 1991; Strang & Fernando +2001). Entrainment rates based on hydrodynamical simulations per- +formed in a stellar context (Meakin & Arnett 2007; Jones et al. 2017; +Cristini et al. 2019) are also implemented in stellar evolution codes to +describe the extension of convective cores and shells (Staritsin 2013; +Scott et al. 2021). However, as shown by Scott et al. (2021), adopting +entrainment rates derived from existing stellar hydrodynamical sim- +ulations to main sequence stellar models produces unrealistic growth +of the convective cores. The parameters that control the entrainment +rates need to be decreased by several orders of magnitude to repro- +duce observations, questioning the reliability of the quantitative rates +derived from existing numerical simulations and even the existence +of an entrainment process for main sequence convective cores. +Describing and isolating these different processes characterising +CBM and at play at convective boundaries can be difficult in numer- +ical simulations. Downward flows (or plumes) crossing a convective +boundary at the bottom of an envelope are clearly observed in nu- +merical simulations (see for example Baraffe et al. 2021). Ballistic +plume crossings may eventually lead to a modification of the thermal +background – the so-called penetration process. But for such modifi- +cation to be observed, simulations must be run over many thousands +of convective turnover timescales, as theoretically expected and re- +cently demonstrated in simulations by Anders et al. (2022) based on +3D simulations of convection in a Cartesian box with idealised se- +tups. In a numerical study of solar-like convective envelopes, Baraffe +et al. (2021) show that artificially boosting the luminosity of the +stellar model by a factor 104 yields a significant modification of +the thermal background below the convective boundary with an ex- +tension of the size of the layer characterised by the penetration of +convective flows, which could lead to a growth of the convectively +unstable zone down to deeper levels. Whether this growth stabilises +or whether the convective boundary continues moving downward +indefinitely is unclear. For the solar-like model with realistic stellar +luminosity, a slight modification of the thermal background is also +observed in the simulations of Baraffe et al. (2021), but they show +no trend of an extension of the Schwarzschild convective boundary +over the simulation time. +Following the approach developed in Pratt et al. (2017) for con- +vective envelopes, the most vigorous plumes can be used to define +a maximal overshooting length, which can be significantly deeper +than the typical length reached by the bulk of the plumes (Pratt et al. +2017; Baraffe et al. 2021; Vlaykov et al. 2022). Whether this bal- +listic process is also observed for convective cores and can drive +significant mixing is an open question. Arguments based on the dy- +namics of convective motions and plumes suggest that mixing below +a convective zone (e.g envelope overshooting) and above (e.g core +overshooting) may indeed be different (Andrássy & Spruit 2013). +Simple arguments based on the kinetic energy of a plume with typ- +ical velocity and the restoring buoyancy force suggest very small +overshooting lengths for the cores of low and intermediate mass +zero-age-main-sequence (ZAMS) stars (Higl et al. 2021). But these +estimates are based on typical velocities without considering possi- +ble extreme plume events. The situation could also be different for +convective cores on the ZAMS and on the main-sequence respec- +tively. Indeed, the building of a molecular weight gradient at the core +boundary due to hydrogen burning in the core can hamper the lifting +of heavier material by ballistic processes. An entrainment process +slowly eroding the convective boundary may thus dominate at some +point over the ballistic process during the main sequence evolution, +or both processes may coexist and contribute to mixing. These ques- +tions are still unsettled. Existing numerical simulations of convective +cores have mostly focussed on one single stellar mass model, rather +than a range of stellar masses (Meakin & Arnett 2007; Gilet et al. +2013; Rogers et al. 2013; Edelmann et al. 2019; Horst et al. 2020; +Higl et al. 2021). Additionally, many of these works enhance the +stellar luminosity of the model, to provide numerical stability, or to +accelerate the thermal relaxation or the Mach number of the con- +vective flow. This artefact may artificially favour one process over +the other. At this time, it is difficult to draw any firm conclusion re- +garding the main mechanisms that drive CBM in stars and how their +efficiency is affected with stellar mass and with the stage of evolution +on the main sequence. +In this work devoted to convective cores, we study the efficiency +for convective plumes to penetrate into the stable region as a function +of stellar mass for ZAMS models. In the following we will refer to +overshooting to describe this process, since we essentially describe +the ballistic process and even if a modification of the temperature +gradient is observed for the most luminous models (see Sect. 5), +likely leading to penetration as defined by Zahn (1991). We perform +two-dimensional (2D) numerical simulations of convective cores of +ZAMS stellar models covering a range of stellar masses between 3 +𝑀⊙ and 20 𝑀⊙ (Sect. 2). Our goal is to apply the framework of ex- +treme plume events developed for convective stellar envelopes (Pratt +et al. 2017, 2020; Baraffe et al. 2021) to the convective cores of inter- +mediate and massive stars. We analyse whether extreme events can +provide overshooting lengths required for stellar models to reproduce +observations. For this purpose, we derive a relationship between over- +shooting length and stellar luminosity based on present numerical +simulations (Sect. 4). We apply the relationship to one-dimensional +stellar evolution models and test them against observations (Sect. 6). +This is the first step for a systematic study devoted to convective core +overshooting in intermediate mass and massive stars. +2 NUMERICAL SIMULATIONS +We use the fully compressible time-implicit code MUSIC. A full +description of MUSIC and of the time-implicit integration can be +found in Viallet et al. (2011, 2016); Goffrey et al. (2017). MUSIC +MNRAS 000, 1–12 (2022) + +A study of convective core overshooting as a function of stellar mass +3 +solves the inviscid Euler equations in the presence of external gravity +and thermal diffusion: +𝜕𝜌 +𝜕𝑡 += +−∇ · (𝜌v), +(1) +𝜕𝜌v +𝜕𝑡 += +−∇ · (𝜌v ⊗ v) − ∇𝑝 + 𝜌g, +(2) +𝜕𝜌𝑒 +𝜕𝑡 += +−∇ · (𝜌𝑒v) − 𝑝∇ · v + ∇ · (𝜒∇𝑇) + 𝑄nuc, +(3) +where 𝜌 is the density, 𝑒 the specific internal energy, v the velocity, +𝑝 the gas pressure, 𝑇 the temperature, g the gravitational accelera- +tion, and 𝜒 the thermal conductivity. The term 𝑄nuc represents the +nuclear energy rate. The symbol ⊗ is the outer product. All hydrody- +namical simulations presented in this work are performed assuming +spherically symmetric gravitational acceleration g, which is updated +every time interval Δ𝑡1. All simulations presented in this work are +performed with Δ𝑡 = 103 s. The typical dynamical timescale of the +entire stellar cores analysed in this study 𝜏dyn ∼ 1/ +√︁ +(𝜌mean𝐺), with +𝜌mean the mean density of the core and 𝐺 the gravitational constant, +is of the order of 103 s. We have checked with a number of test +simulations that a variation of Δ𝑡 between 102 and 105 seconds does +not impact our results. +In the stellar models considered, radiative transfer is the major +heat transport that contributes to the thermal conductivity, which is +given for photons by +𝜒 = 16𝜎𝑇3 +3𝜅𝜌 , +(4) +where 𝜅 is the Rosseland mean opacity, and 𝜎 the Stefan-Boltzmann +constant. Realistic stellar opacities and equation of states appropriate +for the description of stellar interiors are implemented in MUSIC. +Opacities are interpolated from the OPAL tables (Iglesias & Rogers +1996) for solar metallicity and the equation of state is based on the +OPAL EOS tables of Rogers & Nayfonov (2002). +2.1 Initial stellar models +To provide the initial structures for the 2D simulations, we compute +stellar models in the mass range 3-20 𝑀⊙ with the one-dimensional +Lyon stellar evolution code (Baraffe & El Eid 1991; Baraffe et al. +1998), using the same opacities and equation of state as MUSIC2. +The 2D simulations require as initial input a radial profile of den- +sity and internal energy. The 1D stellar evolution models have an +initial helium abundance in mass fraction 𝑌=0.28 and solar metal- +licity 𝑍=0.02 and were computed through the pre-main sequence +and main sequence phases. All initial models for the 2D simulations +in this study are taken at the beginning of core hydrogen burning +and have a central abundance of helium 𝑌c=0.2838, i.e. only ∼ 1% +of their central hydrogen has been depleted. There is thus a very +shallow mean molecular weight gradient at the convective boundary. +Follow-up analysis of later stages of evolution with a steeper gradient +of molecular weight at the core boundary are in progress (Morison +et al. in prep). Convective stability is defined by the Schwarzschild +1 Note that Δ𝑡 is the time after which the gravitational potential is updated, +not the numerical timestep. The numerical timestep used for these simulations +is set by the hydrodynamical CFL number varying between 10 and 50 (see +Viallet et al. 2011, for definitions) and corresponding to values for the timestep +ranging between 5 s and 40 s. +2 The 1D initial structures are available on the repository http://perso.ens- +lyon.fr/isabelle.baraffe/2Dcore_overshooting_2023 +criterion ∇ < ∇ad, with ∇ = d log𝑇 +d log 𝑃 the temperature gradient and +∇ad = d log𝑇 +d log 𝑃 |𝑆 the adiabatic gradient. The 1D stellar models used to +generate the initial structures for the 2D simulations do not account +for overshooting at the convective core boundary. In the following, +we define the Schwarzschild boundary as the transition layer be- +tween convective instability (∇ > ∇ad) and stability (∇ < ∇ad). The +properties of the initial 1D stellar structures are provided in Table 1. +Nuclear energy generated in the convective cores is accounted for in +the internal energy equation (Eq. (3)) through the term 𝑄nuc using +the radial profile of the nuclear energy rate from the 1D stellar model. +Given that the simulation times are orders of magnitude smaller than +the nuclear timescale for H burning in the cores, the nuclear energy +is assumed to remain constant with time. +2.2 Spherical-shell geometry and boundary conditions +Two-dimensional simulations are performed in a spherical shell using +spherical coordinates, namely 𝑟 the radius and 𝜃 the polar angle, and +assuming azimuthal symmetry in the 𝜙-direction. For all models, +the inner radius 𝑟in is defined at 0.02 𝑅star. The choice of the outer +radius 𝑟out depends on the stellar model. Since the main motivation +of this work is to analyse the extent of the overshooting layer for +different stellar masses, the outer radius 𝑟out is fixed at a distance +of ∼ 1 × 𝐻𝑃,CB for the lowest mass (3 𝑀⊙) to ∼ 3.5 × 𝐻𝑃,CB +for the highest mass (20 𝑀⊙) away from the convective boundary +𝑟conv. Extension of the radial domain to analyse the generation of +internal waves at the core boundary and their propagation in the +radiative envelope is work in progress. The angular extent ranges +from 𝜃 = 0◦ to 𝜃 = 180◦. The grid has uniform spacing in the r and +𝜃 coordinates. The choice for the resolution (𝑁𝑟, 𝑁𝜃) is set by the +condition to have a good resolution of the pressure scale height at the +Schwarzschild boundary. Effective Reynolds and Prandtl numbers +are commonly used to set the resolution of numerical simulations. +But given that our simulations are based on an implicit Large Eddy +Simulation (ILES) approach, only a rough estimate can be provided +for these numbers. They will in any case remain far away from the +conditions prevailing in stellar interiors. We suggest that a more +relevant resolution criterion for hydrodynamical simulations devoted +to the study of overshooting using realistic stellar structures should be +the number of grid cells per pressure scale height at the convective +boundary. This should allow a more relevant comparison between +the works of different groups devoted to the study of different stars. +We use ∼ 110 − 140 grid cells per pressure scale height in the +radial direction. The details of the resolution adopted in this work +are provided in Table 2. We have also performed a few tests with +higher resolution and analyse the impact in Sect. 4. +The radial boundary conditions for the density correspond to a +constant radial derivative on the density (see Pratt et al. 2016). The +energy flux at the inner and outer radial boundaries are set to the +value of the energy flux at that radius in the one-dimensional stellar +evolution model. At the boundaries in 𝜃, because of the extension of +the angular domain to the poles, reflective boundary conditions for +the density and energy are used (i.e. the values are mirrored at the +boundary). For the velocity, we impose reflective conditions at the +radial and polar boundaries, corresponding to: +• v𝑟 = 0 and 𝜕v𝜃 +𝜕𝑟 = 0 at 𝑟in and 𝑟out, +• +𝜕v𝑟 +𝜕𝜃 = 0 and v𝜃 = 0 at 𝜃 = 0◦ and 𝜃 = 180◦. +We have also performed simulations for the 3 𝑀⊙ model with +artificial enhancement of the stellar luminosity and the thermal dif- +fusivity by factors 10, 102, 103 and 104. This covers the range of +MNRAS 000, 1–12 (2022) + +4 +I. Baraffe et al. +Table 1. Properties of the initial stellar models (all models have a central helium abundance 𝑌c=0.2838) used for the 2D hydrodynamical simulations: total mass, +stellar luminosity, stellar radius, mass and radius of the convective core (corresponding to the location of the Schwarzschild boundary) and the pressure scale +height at the Schwarzschild boundary. +𝑀/𝑀⊙ +𝐿star/𝐿𝑎 +⊙ +𝑅star (cm) +𝑀conv/𝑀⊙ +𝑟conv/𝑅star +𝐻𝑃,CB (cm) +3 +7.7673 × 101 +1.3855 × 1011 +0.5724 +0.1486 +1.3 × 1010 +5 +5.2186 × 102 +1.8424 × 1011 +1.212 +0.1814 +1.8 × 1010 +10 +5.5726 × 103 +2.7295 × 1011 +3.046 +0.2239 +2.7 × 1010 +15 +1.9242 × 104 +3.4255 × 1011 +5.600 +0.2580 +3.3 × 1010 +20 +4.2962 × 104 +4.0172 × 1011 +8.7947 +0.2869 +3.7 × 1010 +𝑎 We use 𝐿⊙ = 3.839 × 1033 erg/s. +Figure 1. Evolution of the total kinetic energy (in erg; y-axis with a base-10 +log scale) as a function of time (in s) for the simulations described in Tab. +2. Top panel: results for 3 𝑀⊙ models with various luminosity enhancement +factors: 3L0 (black), 3L1 (blue), 3L2 (magenta), 3L3 (cyan) and 3L4 (red). +Bottom panel: results for a range of stellar masses. The dotted line for each +model corresponds to the value of the total kinetic energy at the beginning of +the steady state for convection. +luminosities of the stellar masses considered in this work (3-20 𝑀⊙). +This choice of enhancement factor allows a comparative analysis of +the impact of the luminosity for fixed core mass and increasing core +mass, respectively. Note that even larger enhancement factors (up to +107) for a 3 𝑀⊙ stellar structure can be found in previous works (e.g. +Rogers et al. 2013; Edelmann et al. 2019). For the artificially boosted +simulations, the energy flux (equivalently the luminosity) at the ra- +dial boundaries is multiplied by the enhancement factor, the nuclear +energy rate is multiplied by the same factor and the Rosseland mean +opacities 𝜅 in MUSIC are decreased by the same factor. +Figure 2. Radial profile of the time averaged rms velocity (solid lines) and +rms radial velocity (dashed lines) scaled by (𝐿star/1035)1/3. Top panel: re- +sults for 3 𝑀⊙ models with various luminosity enhancement factors: 3L0 +(black), 3L1 (blue), 3L2 (magenta), 3L3 (cyan) and 3L4 (red). Bottom panel: +results for a range of stellar masses: 3L0 (black), 5L0 (blue), 10L0 (ma- +genta),15L0 (red) and 20L0 (cyan). The convective boundary corresponding +to the Schwarzschild boundary from the 1D initial model is indicated by a +vertical solid line with the colour corresponding to each stellar mass. +3 RESULTS: AVERAGE DYNAMICS +The properties of all simulations are summarised in Table 2. We de- +fine 𝑡steady as the time required to reach a steady state for convection, +characterised by the total kinetic energy 𝐸kin of the system reaching +a plateau. Before 𝑡steady, the initial relaxation phase is characterised +by the propagation of strong acoustic waves and the onset of convec- +tion. At 𝑡steady, the value of the kinetic energy starts to stabilise and +MNRAS 000, 1–12 (2022) + +A study of convective core overshooting as a function of stellar mass +5 +Table 2. Main properties of the 2D simulations. +Model +𝑀/𝑀⊙ +𝐿 (erg/s) +𝑁𝑟 × 𝑁𝜃 +𝑟out/𝑅star +𝜏𝑎conv (s) +𝑁 𝑏 +conv +𝑡𝑐 +steady (s) +𝑡𝑑 +sim (s) +3L0 +3 +2.981 ×1035 +336 x 168 +0.25 +1.9 ×106 +1442 +9.5 ×108 +3.71 ×109 +3L1 +3 +2.981 ×1036 +336 x 168 +0.25 +8 ×105 +1211 +4.6 ×108 +1.43 ×109 +3L2 +3 +2.981 ×1037 +336 x 168 +0.25 +3.9 ×105 +501 +9 ×107 +2.84 ×108 +3L2xhres +3 +2.981 ×1037 +684 x 342 +0.25 +3.8 ×105 +514 +9 ×107 +2.84 ×108 +3L3 +3 +2.981 ×1038 +336 x 168 +0.25 +1.7 ×105 +1904 +6 ×107 +3.81×108 +3L3xhres +3 +2.981 ×1038 +684 x 342 +0.25 +1.7 ×105 +1243 +6 ×107 +2.71 ×108 +3L4 +3 +2.981 ×1039 +336 x 168 +0.25 +8.9 ×104 +1457 +3 ×107 +1.60 ×108 +3L4xhres +3 +2.981 ×1039 +684 x 342 +0.25 +8.7 ×104 +1400 +3 ×107 +1.52 ×108 +5L0 +5 +2.003 ×1036 +400 x 200 +0.3 +1.4 ×106 +1260 +2.45 ×108 +2.01 ×109 +10L0 +10 +2.139 ×1037 +416 x 208 +0.4 +1.2 ×106 +1260 +2.1 × 108 +1.77 ×109 +15L0 +15 +7.387 ×1037 +688 x 344 +0.5 +1.1 ×106 +875 +108 +1.14×109 +20L0 +20 +1.649 ×1038 +864 x 430 +0.6 +1.1 ×106 +800 +9 × 107 +9.99 ×108 +𝑎 Convective turnover time (see Sect. 3 for its definition). +𝑏 Number of convective turnover times covered by the simulation once steady state convection is reached. +𝑐Physical time to reach a steady state for convection. +𝑑Total physical runtime of the simulation. +from this time it remains roughly constant with time (following the +dotted curve which corresponds to the value of 𝐸kin at 𝑡steady for each +model). The simulations are stopped at time 𝑡sim provided in Table 2. +None of these simulations are thermally relaxed, given that the total +simulation times for all models are orders of magnitude smaller than +the relevant thermal timescale ∼ 𝐺𝑀2/(𝑅star𝐿). As a consequence +all these simulations are expected to maintain a secular drift. We +have compared the radial profile of the internal energy, averaged in +the angular direction, for each 2D model at time 𝑡steady and at time +𝑡sim. We find a maximum of 0.5% relative difference for the internal +energy at a given radius, with the largest difference found for the +most luminous models (see Sect. 5). The above-mentioned drift is +thus so slow that calculating statistical or averaged data during this +very slowly changing transitional state is sensible. +Figure 1 shows the evolution of the total kinetic energy as a func- +tion of time for all models and the plateau characterising their steady +state. The initial transient phase can last a relatively long time, de- +pending on the model studied. For the model 3L0, we note a dif- +ferent behaviour. After the peak due to strong acoustic waves, the +kinetic energy continuously decreases until 𝑡 ∼ 2.4 × 108 s (log +𝑡 ∼ 8.38). In this regime, convection develops in the core (within the +1D Schwarzschild boundary) in two spatially separate regions. The +abrupt increase of 𝐸kin observed at 𝑡 ∼ 2.4×108 s marks the merging +of these two convective regions and the beginning of fully developed +convection in the core. The Mach number characterising the con- +vective velocities in model 3L0 is small, of the order of ∼ 10−4, +which is numerically challenging. This low Mach number explains +why several previous works artificially enhance the luminosity of the +model (Rogers et al. 2013; Horst et al. 2020). There is no need for +this artefact for the model 3L0 as MUSIC’s numerical scheme allows +convection to develop and eventually reach a steady state even after a +long transient phase. Note that this unusual transient phase observed +for the model 3L0 will likely change with a different procedure for +initialising the simulation. All simulations start without an imposed +background noise (i.e. initial velocities are set to zero). Imposing +initially a background noise for the model 3L0 may change the loca- +tion where convection starts and thus the behaviour of the transient +phase, which is irrelevant for the analysis performed in the following. +A global convective turnover time 𝜏conv is estimated based on the rms +velocity vrms(𝑟, 𝑡) at radius 𝑟 and time 𝑡, which characterises a bulk +convective velocity. We define 𝜏conv by: +𝜏conv = +�∫ 𝑟conv +𝑟in +d𝑟 +vrms(𝑟, 𝑡) +� +𝑡, +(5) +where the rms velocity is given by +vrms(𝑟, 𝑡) = +√︃ +⟨v2(𝑟, 𝜃, 𝑡)⟩𝜃, +(6) +with v2 = v2𝑟 + v2 +𝜃, v𝑟 and v𝜃 being the radial and angular velocities, +respectively. Time averages are denoted by ⟨⟩𝑡 and calculated between +𝑡steady and 𝑡sim, the final time reached by the simulation (see values +in Table 2). For any quantity 𝑋 we define: +� +𝑋 +� +𝑡 = +1 +(𝑡sim − 𝑡steady) +∫ 𝑡sim +𝑡steady +𝑋d𝑡 +(7) +The volume-weighted average in the angular direction ⟨⟩𝜃 is defined +for any quantity X as: +� +𝑋(𝑟, 𝜃, 𝑡) +� +𝜃 = +∫ +𝜃 𝑋(𝑟, 𝜃, 𝑡)d𝑉(𝑟, 𝜃) +∫ +𝜃 d𝑉(𝑟, 𝜃) +. +(8) +The simulations are stopped after a time 𝑡sim when convergence +of the statistics used to determine the size of the layer penetrated +by plumes is obtained, as explained in the next section (Sect. 4). +Table 2 provides the values and numbers of the convective turnover +times, respectively. Figure 2 displays the rms velocity and rms radial +velocity for the 3 𝑀⊙ models with artificially enhanced luminosities +(upper panel) and for the range of stellar masses investigated (lower +panel), scaled by 𝐿1/3 +star. In the convective core, our simulations re- +produce the expected scaling of convective velocity with luminosity +vconv ∝ 𝐿1/3 recovered by many hydrodynamical simulations (e.g. +Jones et al. 2017; Edelmann et al. 2019; Andrassy et al. 2020; Horst +et al. 2020; Higl et al. 2021; Baraffe et al. 2021). This scaling is +expected from mixing-length theory based on the argument that the +turbulent dissipation rate of kinetic energy in a turbulent convective +zone scales with v3 (Biermann 1932). But a general scaling of the +total flux with v3 can also be derived for the kinetic energy and the +enthalpy fluxes based on simple dimensional arguments (see Jones +et al. 2017) +MNRAS 000, 1–12 (2022) + +6 +I. Baraffe et al. +The rms velocities in the stably stratified region are due to the +penetrative flows just above the convective boundary and to the prop- +agation of internal waves excited by the convective motions and the +penetrating plumes. The top panel of Fig. 2 shows that these ve- +locities also increase with the luminosity, suggesting more efficient +overshooting of the convective motions above the convective bound- +ary and thus larger overshooting length with increasing luminosity. +Baraffe et al. (2021) reports similar behaviours for convective en- +velopes of solar-like models with artificially enhanced luminosities. +Quantitative estimate of the overshooting lengths for all models is +performed in Sect. 4. +4 RESULTS: EXTENT OF THE OVERSHOOTING REGION +4.1 Determination of overshooting lengths +To determine an overshooting length, we adopt the same approach as +in Baraffe et al. (2021) and initially inspired by the findings of Pratt +et al. (2017). This approach is based on the analysis of the depth of +all convective plumes that penetrate beyond the convective boundary. +The two criteria used to determine the depth of a penetrative plume +at a given angle 𝜃 and time 𝑡 are based on the first zero above the +convective boundary 𝑟conv of the vertical kinetic energy flux fk and +vertical heat flux f𝛿T, defined by (see Pratt et al. 2017): +fk(𝑟, 𝜃, 𝑡) = 1 +2 𝜌(𝑟, 𝜃, 𝑡)v2(𝑟, 𝜃, 𝑡)v𝑟 (𝑟, 𝜃, 𝑡), +(9) +f𝛿T(𝑟, 𝜃, 𝑡) = 𝜌(𝑟, 𝜃, 𝑡)𝑐𝑃(𝑟, 𝜃, 𝑡)𝛿𝑇(𝑟, 𝜃, 𝑡)v𝑟 (𝑟, 𝜃, 𝑡), +(10) +where 𝑐𝑃 is the specific heat at constant pressure and the temperature +fluctuation 𝛿𝑇 is defined by: +𝛿𝑇(𝑟, 𝜃, 𝑡) = 𝑇(𝑟, 𝜃, 𝑡) − +�� +𝑇(𝑟, 𝜃, 𝑡) +� +𝜃 +� +𝑡. +(11) +The method is the same as the one developed in Baraffe et al. +(2021) for convective envelopes. At each time 𝑡, we calculate at each +angle 𝜃 the radial positions 𝑟0(𝜃, 𝑡) of a plume corresponding to the +first zero of fk and f𝛿T, respectively, above the convective boundary +𝑟conv. The corresponding overshooting length 𝑙0 with respect to 𝑟conv +is defined by +𝑙0(𝜃, 𝑡) = 𝑟0(𝜃, 𝑡) − 𝑟conv. +(12) +Figure 3 illustrates the angular structure of the overshooting layer at +an arbitrary time for the 10 𝑀⊙ stellar model. +We then define the maximal overshooting length 𝑙max +0 +at a given +time by the maximum over all angles 𝜃: +𝑙max +0 +(𝑡) = max(𝑙0(𝜃, 𝑡)). +(13) +The time average 𝑙max = ⟨𝑙max +0 +(𝑡)⟩𝑡 provides an effective width +for the overshooting layer where the most vigorous plumes penetrate +and which we use to characterise the extension of the mixing layer +over the long term evolution of the star (Pratt et al. 2017; Baraffe +et al. 2021). Table 3 displays 𝑙max based on the criterion for fk and +f𝛿T, respectively, for all models. The distributions of overshooting +lengths derived from fk and f𝛿T, respectively, slowly converges with +time, as found in Pratt et al. (2017) and Baraffe et al. (2021). Several +hundreds to thousand convective turnover times, depending on the +stellar model, are required for the statistics to converge. Eventually, +both criteria provide similar values for the effective overshooting +width. The values of the overshooting width based on f𝛿T converge +faster with time, compared to the value based on fk, as found as +well for convective envelopes in Baraffe et al. (2021). The values +of 𝑙max(f𝛿T) provided in Table 3 have reached a steady state for all +Figure 3. Overshooting lengths𝑙0 defined by Eq. (12) as a function of the angle +𝜃 at time 𝑡 = 8.3108s for the 10 𝑀⊙ model. The upper panel corresponds +to 𝑙0 defined by fk and the lower panel to 𝑙0 defined by f𝛿T. The horizontal +dashed line in each panel indicates the average overshooting length at this +time. +models after 𝑡sim. Depending on the stellar model, 𝑙max(fk) gets close +to 𝑙max(f𝛿T) (difference of <∼ 20%) for all models but models 3L0 +and 20L0, for which 𝑙max(fk) continues slowly decreasing even after +more than 800 ×𝜏conv. We run three simulations for the 3 𝑀⊙ models +with enhanced luminosity with twice the resolution in both radial and +angular directions and covering about the same simulation time as +their lower resolution counterpart, in order to check the sensitivity +of the values of 𝑙max to the resolution. The properties of these higher +resolution models (labelled 2xhres) are displayed in Table 2. The +results for the overshooting lengths are given in Table 3 and show +similar values for lmax(f𝛿T) as found with a lower resolution. The +values for 𝑙max(fk) of the higher resolution models are larger than the +corresponding value for the lower resolution model, as it takes more +time for 𝑙max(fk) in the high resolution models to decrease to the +level of 𝑙max(f𝛿T). But the value of 𝑙max(fk) in the high resolution +models continues decreasing with time and we expect it to eventually +converge and thus get much closer to 𝑙max(f𝛿T) and to the value of +𝑙max(fk) found in the lower resolution model. +4.2 Relationship between overshooting length and stellar +luminosity +The variation of 𝑙max with the stellar luminosity is illustrated in +Fig. 4 for the 3𝑀⊙ models with enhanced luminosity and for the +set of stellar masses with realistic luminosity. As expected from +the behaviour of the rms velocities (see Fig. 2) overshooting lengths +increase with the stellar luminosity. To derive an approximate scaling +relationship for the overshooting length 𝑑ov that can be implemented +in stellar evolution codes, we use the values of 𝑙max derived from f𝛿T, +since these values have converged with time. We derive the following +MNRAS 000, 1–12 (2022) + +A study of convective core overshooting as a function of stellar mass +7 +Table 3. Effective width 𝑙max of the overshooting layer in units of the total stellar radius and of the pressure scale height at the convective boundary, for all +models considered in this study. The quantity 𝑙max(fk) is based on the criterion using fk (Eq. 9) and 𝑙max(f𝛿T) is based on f𝛿T (Eq. 10). +Model +𝑙max(fk)/𝑅star +𝑙max(f𝛿T)/𝑅star +𝑙max(fk)/𝐻𝑃,CB +𝑙max(f𝛿T)/𝐻𝑃,CB +3L0 +6.4 ×10−3 +3.7 ×10−3 +6.8 ×10−2 +3.9 ×10−2 +3L1 +4.2 ×10−3 +4.2 ×10−3 +4.5 × 10−2 +4.5 ×10−2 +3L2 +6.2 ×10−3 +6.1 ×10−3 +6.6 × 10−2 +6.5 ×10−2 +3L2xhres +8.4 ×10−3 +6.4 ×10−3 +8.9 × 10−2 +6.8 ×10−2 +3L3 +1.8 ×10−2 +1.6 ×10−2 +1.9 ×10−1 +1.7 ×10−1 +3L3xhres +2.2 ×10−2 +1.6 ×10−2 +2.3 ×10−1 +1.7 ×10−1 +3L4 +3.5 ×10−2 +2.8 ×10−2 +3.7 ×10−1 +3.0 ×10−1 +3L4xhres +4.0 ×10−2 +3.0 ×10−2 +4.2 ×10−1 +3.2×10−1 +5L0 +9.3 ×10−3 +6.0 ×10−3 +9.5 ×10−2 +6.1 ×10−2 +10L0 +1.2 ×10−2 +1.1 ×10−2 +1.2 ×10−1 +1.1 ×10−1 +15L0 +1.6 ×10−2 +1.3 ×10−2 +1.66 ×10−1 +1.35 ×10−1 +20L0 +3.5 ×10−2 +2.0 ×10−2 +3.8 ×10−1 +2.17 ×10−1 +Figure 4. Overshooting length 𝑙max, in units of the pressure scale height at +the convective boundary, as a function of the model luminosity. The 3 𝑀⊙ +models with various luminosity enhancement factors are indicated in red +(dashed line). The results for a range of stellar masses with realistic stellar +luminosity are indicated in blue (solid line). The dotted curve shows the fit +for the overshooting length 𝑑𝑜𝑣/𝐻P,CB given by Eq. (14). +expression which fits the results for the stellar mass range studied: +𝑑ov/𝐻P,CB = 3.05 × 10−3 × (𝐿/𝐿⊙)1/3 × (𝑟conv/𝐻𝑃,CB)1/2 + 0.02 +(14) +We find a typical scaling with the luminosity 𝑑ov ∝ 𝐿1/3 ∝ vconv. +Numerical studies of convective envelopes report overshooting +lengths 𝑑ov which vary with the luminosity following 𝑑ov ∝ 𝐿𝑎 +with 𝑎 varying between 0.08 and 0.31 (Hotta 2017; Käpylä 2019; +Baraffe et al. 2021). The analytical model of Zahn (1991) for pene- +tration, based on first order estimate of the deceleration of a plume +in an adiabatically stratified penetration zone, predicts 𝑑ov ∝ v3/2 +conv. +Our results also show that the overshooting lengths derived for a fixed +stellar mass (and thus a fixed convective core size) are systematically +smaller than the one derived for larger cores but similar luminosity. +Interestingly, a dependence of 𝑑ov with the size of the core 𝑟conv is +also predicted by Zahn (1991) (see their Eq. (4.5)) with the same re- +lation of proportionality 𝑑ov ∝ (𝑟conv/𝐻𝑃,CB)1/2 as found in present +simulations. This dependence in the Zahn model is derived from the +strong variations with radius of various relevant quantities such as +the gravitational acceleration 𝑔, the mass 𝑚(𝑟) enclosed in a sphere +of radius 𝑟, the radiative conductivity 𝜒, and thus the radiative flux, +close to the convective core boundary. In our simulations, we expect +the radial dependence of the gravitational acceleration to have the +main impact. We find that the larger the core (in terms of radius and +mass), the smaller the gravitational acceleration at the core boundary +𝑔conv ∼ 𝐺𝑀conv/𝑟2conv (see values in Table 1). Therefore, the larger +the stellar mass, the larger the velocities at the convective boundary +and the smaller the restoring force due to gravity, implying up-flows +to penetrate over larger distances. This is a plausible explanation for +the dependence of 𝑑ov on the convective core radius. We analyse +below (Sect. 6) whether the expression provided by Eq. (14) pro- +vides a reasonable agreement between stellar evolution models and +observations. +5 THERMAL BACKGROUND EVOLUTION +The prescription used in the previous section to determine overshoot- +ing lengths relies on two assumptions. Firstly, we consider that the +simulations have reached a steady state for convection (i.e. a global +dynamical steady state). This assumption is reasonable based on +the observation that the total kinetic energy of the system reaches +a plateau as a function of time (see Fig. 1). Secondly, we assume +that the relevant convective boundary from which the overshooting +lengths are defined is the 1D Schwarzschild boundary. This is directly +useful for the purpose of implementing these overshooting lengths +in 1D stellar evolution codes. However, we find that in all models +a small nearly adiabatic layer just above the convective boundary +forms rapidly once convection steady state is reached. For the most +luminous models, we observe that this small layer slowly grows in +size with time. +Anders et al. (2022) also find a modification of the temperature +gradient which becomes close to the adiabatic gradient in the pen- +etration layer. They report that their simulations exhibit the process +of convective penetration as defined by e.g. Zahn (1991), with con- +vective penetrating motions mixing entropy and establishing a nearly +adiabatic stratification above the Schwarzschild boundary (see also +Brummell et al. 2002). Anders et al. (2022) suggest that the extent of +convective penetration is limited and derive arguments involving the +MNRAS 000, 1–12 (2022) + +8 +I. Baraffe et al. +. +Figure 5. Visualisation of the radial velocity v𝑟 [cm/s] (top panel) and the +relative temperature fluctuations (𝑇 −⟨𝑇 ⟩𝜃)/⟨𝑇 ⟩𝜃 (bottom panel) in a region +zoomed around the convective boundary (horizontal black line) for the model +20L0 at time 𝑡 = 7 × 108 s. The x-axis represents the co-latitude (in terms of +cos 𝜃). Note that to the better illustrate upwellings and downwellings in the +top panel, the velocity scale is saturated, i.e. any velocity > vr,max = 5 × 103 +cm/s (< vr,min = −5×103 cm/s) are represented with the same color as vr,max +(vr,min). +convective flux, the viscous dissipation rate and the buoyancy work, +providing an estimate of the penetration width. Depending on their +setup, they find that penetration zones can take thousands of con- +vective turnover times to saturate. They show properties of the flow +and of the temperature fluctuations close to a convective boundary +(see their Figure 1) which are similar to our results, as illustrated +in Fig. 5 for the model 20L0 at a given time. As expected in con- +vective regions, convective upflows transport hot material from the +central regions up to the top of the convective core. Inspection of +temperature fluctuations (i.e. the difference between the local tem- +perature and the horizontally averaged thermal background) indeed +indicates that upflows in the convective region are characterised by +positive temperature fluctuations and downflows by negative tem- +perature fluctuations. When upflows cross the convective boundary, +at the top of the convective core, and penetrate the stably stratified +medium, they adiabatically expand and therefore get cooler (neg- +ative temperature fluctuation) and denser than the subadiabatically +stratified environment. +To understand the establishment of a nearly adiabatic layer in the +penetration region, one needs to compare the advection timescale, +which characterises the process of entropy mixing by penetrating +flows (i.e. an advection process), and the thermal diffusion timescale. +If penetrating flows, as illustrated in the top panel of Fig. 5, can drive +efficient entropy/thermal mixing, the layer characterised by pene- +trating up-flows will remain nearly adiabatic if thermal diffusion is +slow enough. Table 4 provides estimates of the diffusive timescale +Figure 6. Profile of the time and angular averages of the quantity (∇ − ∇ad) +in the layers just above the convective core for the most luminous models. +The 1D profile of (∇ − ∇ad) is indicated by the black dashed line and the +1D convective core boundary by the vertical solid line. The location of 𝑙max +derived from f𝛿T is indicated by the vertical dashed line. In both panels, the +solid blue line corresponds to the time average between 𝑡steady and 𝑡sim. The +curves in magenta correspond to time averages over 20×𝜏conv at a given time, +as indicated in each panel (time 𝑡 in s). +𝜏diff = 𝐿2/𝜅rad at the core boundary, with 𝐿 a relevant lengthscale +and 𝜅rad = 𝜒/(𝜌𝑐𝑃) the thermal diffusivity (which is the radiative +diffusivity for present stellar models with 𝜒 defined in Eq. (4)). Esti- +mate of an advection timescale 𝜏adv = 𝐿/vr,rms is based on the time +averaged rms radial velocity at the core boundary. For the charac- +teristic lengthscales at the core boundary, we use the overshooting +distance 𝑙max(f𝛿T) (see Table 3) and the pressure scale height 𝐻P +(see Table 1). As illustrated in Table 4, typical advection timescales +are much smaller than typical thermal diffusion timescales for all +models. +The growth in size with time of the nearly adiabatic layer observed +in the most luminous models is illustrated in Fig. 6 for the models +3L3 and 3L4. This growth with time may also happen in the less +luminous models, but their very slow evolution and less vigorous +penetrating flows may prevent clearly exhibiting this feature over +present simulation times. We also note that the angular averaged +temperature gradient in the models, while getting very close to the +adiabatic gradient, remains stable against the Schwarzschild criterion +over the simulation times. +For the purpose of analysing the time evolution of the nearly +adiabatic layer, we have extended the simulation time of the models +3L3 and 3L4 beyond the value of 𝑡sim used to determine overshooting +depths (see Tab. 2), until 𝑡final = 5 × 108 s (∼ 2600 × 𝜏conv for 3L3 +and ∼ 5300×𝜏conv for 3L4). The aim is to reach a simulation time for +these models close to or greater than the thermal diffusion timescale +in the overshooting layer 𝜏diff(𝑙max). Given the smaller grid size and +larger thermal diffusivity of these models, this is still computationally +affordable. Figure 6 shows clearly in models 3L3 and 3L4 that the +radial extension of the nearly adiabatic layer slows down with time +MNRAS 000, 1–12 (2022) + +0.36 +4000 +0.34 +0.32 +2000 +0.30 +0 +0.28 +0.26 +-2000 +0.24 +-4000 +0.22 +-1.00 -0.75 -0.50 -0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +cos θ0.36 +10-3 +0.34 +10-4 +0.32 +10-5 +10-6 +0.30 +T- (T)e)/∼ 15𝑀⊙. We use this catalog of data to test the scaling +relationship for 𝑑ov predicted by present numerical simulations. +Stellar evolution models are calculated using the MESA code (Pax- +ton et al. 2011) which provides the flexibility of easily implementing +MNRAS 000, 1–12 (2022) + +3.8 - +3.6 +3.4 +(°/)60| +3.2 +Observed +3.0 - +8Mo +9Mo +10Mo +2.8 +12Mo +15Mo +20Mo +Best Fit +2.6 - +No Overshooting +4.7 +4.6 +4.5 +4.4 +4.3 +4.2 +4.1 +log(Teff)(k)10 +I. Baraffe et al. +Table 5. Values of 𝑑ov/𝐻P for each stellar model evolved with the scaling +relationship given by Eq. (14) at the ZAMS and the TAMS, respectively. 𝛼ov +is the fitted value for each stellar mass required to roughly reproduce the +observed main sequence width. +𝑀/𝑀⊙ +𝑑ov/𝐻P (ZAMS) +𝑑ov/𝐻P (TAMS) +Fitted 𝛼ov +8 +0.09 +0.11 +0.1 +9 +0.10 +0.13 +0.2 +10 +0.11 +0.14 +0.3 +12 +0.13 +0.18 +0.35 +15 +0.16 +0.23 +0.4 +20 +0.22 +0.32 +0.45 +the scaling relation for overshooting distance given by Eq. (14). +Instantaneous mixing is assumed over the distance 𝑑ov above the +convective core. We have performed calculations adopting either a +radiative or an adiabatic temperature gradient in the overshooting +layer and find no significant impact on the evolutionary tracks. As +done in Castro et al. (2014), we compare the data to solar metal- +licity models. In Fig. 7 we show the evolution of massive stars in +the mass range 8-20 𝑀⊙ with no overshooting and with the scaling +relationship given by Eq. (14). The tracks are compared to the Castro +et al. (2014) data and to the empirical locations of the ZAMS and +the TAMS. Table 5 provides the values of 𝑑ov/𝐻P at the ZAMS +and the TAMS, respectively, for the models evolved with the scaling +relationship given by Eq. (14). +We have also computed models with an arbitrary overshooting +length 𝑑ov = 𝛼ov𝐻P which is fixed for a given stellar mass but +increases with mass. The values of 𝛼ov for this set of models are +chosen to roughly reproduce the main sequence width and are pro- +vided in Table 5. We did not try to reproduce the ZAMS/TAMS +empirical positions accurately. This set of models is also shown in +Fig. 7. In agreement with the conclusions of Castro et al. (2014), +models without overshooting are unable to reproduce the observed +main sequence width. An increasing overshooting distance with in- +creasing stellar mass is required to reproduce the observed width. +The overshooting scaling law based on our present hydrodynami- +cal simulations predict this increase with the stellar mass (see Fig. +4). It provides a good fit to the observed main sequence width for +𝑀 <∼ 10𝑀⊙. But it tends to under-predict the value of 𝑑ov needed for +models of higher mass to reach the observed location of the TAMS. +A comparison of the values of 𝑑ov given in Table 5 with the fitted +values of 𝛼ov given in the same table suggests that values predicted +by the hydrodynamical simulations are a factor ∼ 2 smaller than what +is required to reach the observed location of the TAMS. +6.2 Massive binaries +We also test the overshooting scaling law given by Eq. (14) against +a selected sample of massive eccentric binaries, namely HD 152218 +(Rauw et al. 2016), HD152219 (Rosu et al. 2022b) and CPD-41◦742 +(Rosu et al. 2022a). We limit present analysis to this restricted number +of binary systems as they belong to the same young open cluster NGC +6231 and thus have the same metallicity, likely a solar metallicity. +In addition, their fundamental properties are inferred using the same +methods and tools. This selected sample thus provides a small but +homogeneous and consistent set of data to compare to stellar models. +Their fundamental properties are provided in Table 6. +Figure 8 compares evolutionary tracks with different treatments of +core overshooting with the observed properties of these binaries. For +Table 6. Properties of the binaries used for the comparison with models. +Binary +𝑀/𝑀⊙ +𝑇eff(K) +𝐿/𝐿⊙ +HD 152218a +19.8 ±1.5 +33 400 ±1000 +7.94+2.52 +−1.77 × 104 +HD 152218b +15.0 ±1.1 +29 900 ±1000 +4.36+1.39 +−1.48 × 104 +HD 152219a +18.64 ±0.47 +30 900 ±1000 +(7.26±0.97) × 104 +HD 152219b +7.70 ±0.12 +21 697 ±1000 +(2.73±0.51) × 103 +CPD-41◦742a +17.8 ±0.5 +31 800 ±1000 +5.28+0.67 +−0.68 × 104 +CPD-41◦742b +10.0 ±0.3 +24 098 ±1000 +5.58+0.93 +−0.94 × 103 +HD 152218 (Fig. 8, left panel), all models provide a solution within +the error bars, but the large error bars do not provide a very stringent +test for the treatment of overshooting. For HD 152219 (Fig. 8, middle +panel), all models provide a solution to the secondary, while only +models with the arbitrary overshooting width from Tab. 5 provide a +solution for the primary. Finally, for CPD-41◦742 (Fig. 8, right panel), +all models fall within the error bars for the secondary. For the primary, +models with the arbitrary overshooting width provide a solution, but +the models with the present hydrodynamical relationship provide +solutions at the very limit of the error bars. Although this comparison +of models with binaries is less conclusive than the one performed +with the Castro et al. (2014) data, it suggests that larger overshooting +widths than predicted by the hydrodynamical relationship would +provide a better fit, particularly for primaries with masses ∼ 18 𝑀⊙. +7 DISCUSSION AND CONCLUSION +This work is an initial, exploratory investigation, in which we infer +an overshooting width 𝑑ov for a broad range of ZAMS stellar models +based on hydrodynamical simulations. The present determination of +an effective overshooting width, characterising the extent of mixing +on the long term evolution of the star, is based on an approach rely- +ing on extreme events of penetrating flows previously developed for +convective envelopes of solar-type stars (e.g. Pratt et al. 2017, 2020; +Baraffe et al. 2021). For ZAMS stars, we find that the overshoot- +ing distance scales with the stellar luminosity and the convective +core radius, resulting in values of 𝑑ov which significantly increase +with stellar mass. Obtaining this increase is an important achieve- +ment, since such an increase is suggested by several observational +constraints. But although the results within our framework are qual- +itatively in agreement with the observed trends, quantitatively, they +are unable to match the available data. Indeed, the comparison of +stellar evolution tracks to the properties of a sample of Milky Way +main sequence stars suggests that the predicted values of 𝑑ov are +underestimated for 𝑀 >∼ 10𝑀⊙. The comparison to massive bina- +ries suggests the same limitation. This points to a need for further +computational studies, as discussed below. +The diagnostics we have used to examine the present set of 2D +simulations have their limitations and several physical or numeri- +cal ingredients may increase the values of overshooting lengths. One +limitation is our assumption that the overshooting lengths determined +within the extreme event framework, which is based on fluxes in an +Eulerian approach, characterise the extension of efficient chemical +mixing above the convective core. Quantifying the extent of chemical +mixing is the prime interest for an application to 1D stellar evolu- +tion models. This assumption can be verified with an analysis of +mixing based on Lagrangian tracer particles, a direction that will +MNRAS 000, 1–12 (2022) + +A study of convective core overshooting as a function of stellar mass +11 +Figure 8. Comparison of evolutionary tracks with different treatments of overshooting and observations for massive binaries in the Hertzsprung-Russell diagram. +Green lines: Models evolved with the overshooting law given by Eq. (14). Red lines: models evolved with an arbitrary overshooting length 𝑑ov provided in +Table 5 (Fittted 𝛼ov). Blue lines: models without overshooting. The solid lines correspond to the track for the masses provided in Table 6 and the dashed lines +correspond to the tracks for the upper and lower masses within the errorbars. Observations are from Rauw et al. (2016) for HD 152218, Rosu et al. (2022b) for +HD152219 and Rosu et al. (2022a) for CPD-41◦742. +be explored in a future work. The formation of a small nearly adia- +batic layer above the convective core of our models due to efficient +entropy mixing by the upward penetrating flows indicates that effi- +cient chemical mixing should also proceed between the convective +boundary and the location of the maximal overshooting length 𝑙max. +But the size of the layer for efficient chemical mixing and the one +of the nearly adiabatic layer are not expected to be the same, even +if the same initial process drives thermal and chemical mixing (i.e. +advection by upward flows). Our results suggest that the extent of +the nearly adiabatic layer may be limited by thermal diffusion, as +observed for the most luminous models when the simulation time +exceeds the typical thermal diffusive timescale in the overshooting +layer. Thermal diffusion will not limit the extent of chemical mixing. +Internal waves excited by convective plumes at the core boundary +could however contribute to additional chemical mixing and extend +the size of the chemical mixing layer beyond 𝑙max. This is also un- +der further investigation and could provide an interesting process to +increase the overshooting lengths derived with present approach. +Extension to three-dimensional geometry is an obvious next step, +since the structure and the geometry of penetrating convective flows +are expected to be modified in 3D compared to 2D simulations (see +Brummell et al. 2002). Despite 2D convective velocities being on +average larger than 3D velocities (Meakin & Arnett 2007; Pratt et al. +2020), several works have suggested that the filling factor and plume +geometry could be smaller in 3D than in 2D (see discussion in Rogers +et al. 2006). Simulations in 3D may thus provide larger overshooting +lengths, as needed to reproduce stellar observations. But so far no +conclusive study of the filling factor and plume shape using the same +simulation framework in 2D and 3D has been performed (see Pratt +et al. 2020). +Further numerical studies need to be performed in order to de- +termine the impact of rotation and whether it can provide another +driver to increase overshooting lengths and/or to make mixing more +efficient (see e.g. Browning et al. 2004). A limitation of the present +simulations, and indeed many global simulations of stars, is the fact +that they are not thermally relaxed, since this would require simu- +lation times even greater than the values for the thermal diffusion +timescale over a pressure scale height 𝜏diff(𝐻P) provided in Table 4. +The direct application of the overshooting lengths predicted by these +simulations to “real" stars must thus be taken with caution, since the +final relaxed state for these simulations may have different properties +from present non thermally relaxed states. This does not however pre- +clude analysing the efficiency of overshooting as a function of stellar +mass and luminosity during the slowly evolving transient phase dur- +ing which convection is considered to be in steady state. One can +speculate that even if the convective boundary moves with respect to +the initial 1D Schwarzschild boundary after thermal relaxation, the +overshooting lengths determined on a dynamical steady state from +MNRAS 000, 1–12 (2022) + +HD152219 +5.00 ++ +4.75 +4.50 +(7/7)60l +4.25 +4.00 +3.75 +3.50 +4.60 +4.55 +4.504.454.404.354.30 +4.25 +4.20 +log(Teff) (k)5.0 +CPD-410 742 +4.8 +4.6 +(7/7)60l +4.4 +4.2 +4.0 +3.8 +3.6 +4.60 +4.55 +4.50 +4.454.40 +4.35 +4.30 +4.25 +4.20 +log(Teff) (k)5.2 +HD152218 +5.0 - +4.8 +(7/7)60| +4.6 +4.4 +4.2 +4.60 +4.55 +4.50 +4.454.40 +4.35 +4.30 +4.25 +4.20 +log(Teff) (k)12 +I. Baraffe et al. +this new boundary may still be close to the the ones determined in +this work. Unfortunately, to verify this implies running the simula- +tions over a thermal timescale, which is computationally not feasible. +More extreme enhancement factors for the luminosity could allow +reaching thermal relaxation. But as shown recently for convective en- +velopes in Baraffe et al. (2021), large enhancement factors can push +the simulated conditions away from the original target star, inducing +a significant drift from the initial stellar structure. +In addition, the scaling presented in this work is derived for ZAMS +stars and may not apply to cores that have evolved on the main +sequence. Indeed, the development of a molecular weight gradient +at the core boundary due to hydrogen burning will most likely limit +the radial penetration of upward flows above the convective core +boundary. Numerical simulations of the convective core of main +sequence 5 𝑀⊙ and 20 𝑀⊙ star models indicate much smaller values +of 𝑙max compared to their ZAMS counterpart (Morison et al., in prep). +In addition, they show no sign of entrainment which could result in +an increase of the size of the convective core. Whether 3D, rotation +and/or other instabilities can solve the problem of “impenetrability" +of convective flows due to the building of a molecular weight gradient +during the evolution on the main sequence is an open question. Other +effects and/or improvement of present 2D simulations are needed to +increase the overshooting lengths for both ZAMS and main sequence +models. +In conclusion, this work provides results which qualitatively vali- +date the increase of overshooting lengths with stellar mass (or stellar +luminosity) suggested by observations (e.g. Castro et al. 2014). Quan- +titatively, however, the predicted values are underestimated for stellar +masses >∼ 10𝑀⊙. Our present results apply only to stellar models on +the ZAMS. Our study illustrates the challenges and the promise of +hydrodynamical simulations. It sets the stage for broader, and more +physically detailed studies to resolve in the future this quantitative +discrepancy with observations. +ACKNOWLEDGEMENTS +This work is supported by the ERC grant No. 787361-COBOM +and the consolidated STFC grant ST/R000395/1. We are grateful +to Noberto Castro for providing data in a user friendly form and +for useful advises for using the catalog. We thank our anonymous +referee for very valuable comments and suggestions. The authors +would like to acknowledge the use of the University of Exeter High- +Performance Computing (HPC) facility ISCA and of the DiRAC +Data Intensive service at Leicester, operated by the University of +Leicester IT Services, which forms part of the STFC DiRAC HPC +Facility. The equipment was funded by BEIS capital funding via +STFC capital grants ST/K000373/1 and ST/R002363/1 and STFC +DiRAC Operations grant ST/R001014/1. DiRAC is part of the Na- +tional e-Infrastructure. Part of this work was performed under the +auspices of the U.S. Department of Energy by Lawrence Livermore +National Laboratory under Contract DE-AC52-07NA27344. +DATA AVAILABILITY +The 1D initial structures are available on the repository: +http://perso.ens-lyon.fr/isabelle.baraffe/2Dcore_overshooting_2023. +The other data underlying this article will be shared on reasonable +request to the corresponding author. +REFERENCES +Anders E. H., Jermyn A. S., Lecoanet D., Brown B. P., 2022, ApJ, 926, 169 +Andrássy R., Spruit H. C., 2013, A&A, 559, A122 +Andrassy R., Herwig F., Woodward P., Ritter C., 2020, MNRAS, 491, 972 +Arnett W. D., Meakin C., 2011, ApJ, 733, 78 +Baraffe I., El Eid M. F., 1991, A&A, 245, 548 +Baraffe I., Chabrier G., Allard F., Hauschildt P. H., 1998, A&A, 337, 403 +Baraffe I., Pratt J., Vlaykov D. G., Guillet T., Goffrey T., Le Saux A., Con- +stantino T., 2021, A&A, 654, A126 +Biermann L., 1932, Z. Astrophys., 5, 117 +Bossini D., et al., 2015, MNRAS, 453, 2290 +Browning M. K., Brun A. S., Toomre J., 2004, ApJ, 601, 512 +Brummell N. H., Clune T. L., Toomre J., 2002, ApJ, 570, 825 +Castro N., Fossati L., Langer N., Simón-Díaz S., Schneider F. R. N., Izzard +R. G., 2014, A&A, 570, L13 +Claret A., Torres G., 2016, A&A, 592, A15 +Claret A., Torres G., 2019, ApJ, 876, 134 +Cristini A., Hirschi R., Meakin C., Arnett D., Georgy C., Walkington I., 2019, +MNRAS, 484, 4645 +Edelmann P. V. F., Ratnasingam R. P., Pedersen M. G., Bowman D. M., Prat +V., Rogers T. M., 2019, ApJ, 876, 4 +Fernando H. J. S., 1991, Annual Review of Fluid Mechanics, 23, 455 +Freytag B., Ludwig H. G., Steffen M., 1996, A&A, 313, 497 +Gilet C., Almgren A. S., Bell J. B., Nonaka A., Woosley S. E., Zingale M., +2013, ApJ, 773, 137 +Goffrey T., et al., 2017, A&A, 600, A7 +Goldreich P., Kumar P., 1990, ApJ, 363, 694 +Higl J., Müller E., Weiss A., 2021, A&A, 646, A133 +Horst L., Edelmann P. V. F., Andrássy R., Röpke F. K., Bowman D. M., Aerts +C., Ratnasingam R. P., 2020, A&A, 641, A18 +Hotta H., 2017, ApJ, 843, 52 +Iglesias C. A., Rogers F. J., 1996, ApJ, 464, 943 +Johnston C., 2021, A&A, 655, A29 +Jones S., Andrassy R., Sandalski S., Davis A., Woodward P., Herwig F., 2017, +MNRAS, 465, 2991 +Käpylä P. J., 2019, A&A, 631, A122 +Lecoanet D., Quataert E., 2013, MNRAS, 430, 2363 +Meakin C. A., Arnett D., 2007, ApJ, 667, 448 +Michielsen M., Pedersen M. G., Augustson K. C., Mathis S., Aerts C., 2019, +A&A, 628, A76 +Montalbán J., Schatzman E., 2000, A&A, 354, 943 +Paxton B., Bildsten L., Dotter A., Herwig F., Lesaffre P., Timmes F., 2011, +ApJS, 192, 3 +Pinçon C., Belkacem K., Goupil M. J., 2016, A&A, 588, A122 +Pratt J., et al., 2016, A&A, 593, A121 +Pratt J., Baraffe I., Goffrey T., Constantino T., Viallet M., Popov M. V., Walder +R., Folini D., 2017, A&A, 604, A125 +Pratt J., Baraffe I., Goffrey T., Geroux C., Constantino T., Folini D., Walder +R., 2020, A&A, 638, A15 +Press W. H., 1981, ApJ, 245, 286 +Rauw G., Rosu S., Noels A., Mahy L., Schmitt J. H. M. M., Godart M., Dupret +M. A., Gosset E., 2016, A&A, 594, A33 +Rieutord M., Zahn J. P., 1995, A&A, 296, 127 +Rogers F. J., Nayfonov A., 2002, ApJ, 576, 1064 +Rogers T. M., Glatzmaier G. A., Jones C. A., 2006, ApJ, 653, 765 +Rogers T. M., Lin D. N. C., McElwaine J. N., Lau H. H. B., 2013, ApJ, 772, +21 +Rosenfield P., et al., 2017, ApJ, 841, 69 +Rosu S., Rauw G., Nazé Y., Gosset E., Sterken C., 2022a, arXiv e-prints, p. +arXiv:2205.11207 +Rosu S., Rauw G., Farnir M., Dupret M. A., Noels A., 2022b, A&A, 660, +A120 +Schatzman E., 1993, A&A, 279, 431 +Scott L. J. A., Hirschi R., Georgy C., Arnett W. D., Meakin C., Kaiser E. A., +Ekström S., Yusof N., 2021, MNRAS, 503, 4208 +Shaviv G., Salpeter E. E., 1973, ApJ, 184, 191 +MNRAS 000, 1–12 (2022) + +A study of convective core overshooting as a function of stellar mass +13 +Stancliffe R. J., Fossati L., Passy J. C., Schneider F. R. N., 2016, A&A, 586, +A119 +Staritsin E. I., 2013, Astronomy Reports, 57, 380 +Strang E. J., Fernando H. J. S., 2001, Journal of Fluid Mechanics, 428, 349 +Viallet M., Baraffe I., Walder R., 2011, A&A, 531, A86 +Viallet M., Goffrey T., Baraffe I., Folini D., Geroux C., Popov M. V., Pratt J., +Walder R., 2016, A&A, 586, A153 +Vlaykov D. G., Baraffe I., Constantino T., Goffrey T., Guillet T., Le Saux A., +Morison A., Pratt J., 2022, MNRAS, 514, 715 +Zahn J. P., 1991, A&A, 252, 179 +Zahn J. P., 1992, A&A, 265, 115 +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–12 (2022) + diff --git a/1tE0T4oBgHgl3EQfuQHv/content/tmp_files/load_file.txt b/1tE0T4oBgHgl3EQfuQHv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..28e63506ba33dab56641e045dafbca69c288109a --- /dev/null +++ b/1tE0T4oBgHgl3EQfuQHv/content/tmp_files/load_file.txt @@ -0,0 +1,1235 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf,len=1234 +page_content='MNRAS 000, 1–12 (2022) Preprint 9 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='0 A study of convective core overshooting as a function of stellar mass based on two-dimensional hydrodynamical simulations I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Baraffe,1,2 ★ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Clarke,1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Morison,1 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Vlaykov,1 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Constantino,1 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Goffrey,3 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Guillet,1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Le Saux1,2 and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Pratt4 1University of Exeter, Physics and Astronomy, EX4 4QL Exeter, UK 2École Normale Supérieure, Lyon, CRAL (UMR CNRS 5574), Université de Lyon, France 3Centre for Fusion, Space and Astrophysics, Department of Physics, University of Warwick, Coventry, CV4 7AL, UK 4Lawrence Livermore National Laboratory, 7000 East Ave, Livermore, CA 94550, USA Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Received YYY ABSTRACT We perform two-dimensional numerical simulations of core convection for zero-age-main-sequence stars covering a mass range from 3 𝑀⊙ to 20 𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The simulations are performed with the fully compressible time-implicit code MUSIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We study the efficiency of overshooting, which describes the ballistic process of convective flows crossing a convective boundary, as a function of stellar mass and luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We also study the impact of artificially increasing the stellar luminosity for 3 𝑀⊙ models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The simulations cover hundreds to thousands of convective turnover timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Applying the framework of extreme plume events previously developed for convective envelopes, we derive overshooting lengths as a function of stellar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We find that the overshooting distance (𝑑ov) scales with the stellar luminosity (𝐿) and the convective core radius (𝑟conv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We derive a scaling law 𝑑ov ∝ 𝐿1/3𝑟1/2 conv which is implemented in a 1D stellar evolution code and the resulting stellar models are compared to observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The scaling predicts values for the overshooting distance that significantly increase with stellar mass, in qualitative agreement with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Quantitatively, however, the predicted values are underestimated for masses >∼ 10𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Our 2D simulations show the formation of a nearly-adiabatic layer just above the Schwarzschild boundary of the convective core, as exhibited in recent 3D simulations of convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The most luminous models show a growth in size with time of the nearly-adiabatic layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This growth seems to slow down as the upper edge of the nearly-adiabatic layer gets closer to the maximum overshooting length and as the simulation time exceeds the typical thermal diffusive timescale in the overshooting layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Key words: Convection – Hydrodynamics – Stars: evolution 1 INTRODUCTION One of the major uncertainties in stellar evolution models is the treat- ment of mixing taking place at convective boundaries (see Stancliffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Convective motions do not abruptly stop at the classical Schwarzschild boundary, but extend beyond it and lead to the pro- cess of convective boundary mixing (CBM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The complex dynamics resulting from convective flows penetrating in stable layers drives the transport of chemical species and heat, strongly affecting the structure and the evolution of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The same complex dynamics can also drive transport of angular momentum, impacting the rotational evolution of stars, the generation of magnetic field in their interior and their magnetic activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' CBM affects the evolution of all stars that develop a convective envelope, core or shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Its treatment is one of the oldest unsolved problems of stellar structure and evolution theory (Shaviv & Salpeter 1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This extra mixing could significantly alter the size of a convective core, the lifetime of major burning phases or the surface chemistry over a wide range of stellar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' It can impact the entire evolution of massive stars (𝑀 >∼ 8𝑀⊙), determin- ing their structure before core-collapse supernova explosion and thus ★ E-mail: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='baraffe@ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='uk affecting nucleosynthetic yields which are crucial for galactic evolu- tion studies (Arnett & Meakin 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' There is ample observational evidence pointing towards the need for extra internal mixing to ex- plain a wide range of observations, such as eclipsing binaries (Claret & Torres 2016), color-magnitude diagrams (Rosenfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2017) or asteroseismology (Bossini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Rosenfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2017) illustrate the uncertainty due to the treatment of core overshooting on ages and on morphological changes in stellar evolution tracks, signif- icantly impacting stellar population studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' An increasing number of observational studies also suggests an increase of convective bound- ary mixing efficiency with stellar mass, using eclipsing binaries (see Claret & Torres 2019, and references therein) or Hertzsprung-Russell diagrams of massive stars (Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' In a recent study, John- ston (2021) confirms that current stellar models with no or with little convective boundary mixing usually under-predict the mass of con- vective cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' While such comparisons between stellar models and observations cannot identify a mechanism responsible for mixing at the convective boundaries, Johnston (2021) concludes that a range of efficiencies for the mixing mechanism(s) should be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' In addition to CBM, additional mixing could be due to rotation (Zahn 1992) or internal gravity waves (Schatzman 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The latter are connected to CBM as they are excited at convective boundaries by turbulent con- © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='02604v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='SR] 6 Jan 2023 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' vective motions (Press 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Goldreich & Kumar 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Lecoanet & Quataert 2013) and penetrating flows (Rieutord & Zahn 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Montalbán & Schatzman 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Pinçon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' CBM is a generic term that encompasses different processes, namely penetration, overshooting or entrainment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The first term de- scribes motions that cross a convective boundary and alter the back- ground in such a way that the location of the convective boundary, defined by the Schwarzschild or the Ledoux criterion, moves inward or outward, resulting in the extension of the convective region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Over- shooting usually describes convective penetrative motions that do not alter the background but can still result in more or less efficient mixing (Zahn 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' In the literature, the terms overshooting and penetration are often used interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' These processes have been described in stellar evolution models by an overshooting distance 𝑑ov and/or a diffusion coefficient which remains constant or exponentially decays over the overshooting length (Freytag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' These parameters are usually calibrated to fit observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The temperature gradient in the overshooting region is either set to the radiative or to the adiabatic temperature gradient (see for example Michielsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The third term entrainment is used to characterise shear-induced turbulent motions at the interface between the convectively stable and unstable regions driven by convective penetrative motions (plumes or eddies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Interfacial instabilities contribute to mixing fluids of different com- positions and/or densities, eroding the convective boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This one can then grow in time following an entrainment rate characterised by the bulk Richardson number (Fernando 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Strang & Fernando 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Entrainment rates based on hydrodynamical simulations per- formed in a stellar context (Meakin & Arnett 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Cristini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2019) are also implemented in stellar evolution codes to describe the extension of convective cores and shells (Staritsin 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Scott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' However, as shown by Scott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2021), adopting entrainment rates derived from existing stellar hydrodynamical sim- ulations to main sequence stellar models produces unrealistic growth of the convective cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The parameters that control the entrainment rates need to be decreased by several orders of magnitude to repro- duce observations, questioning the reliability of the quantitative rates derived from existing numerical simulations and even the existence of an entrainment process for main sequence convective cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Describing and isolating these different processes characterising CBM and at play at convective boundaries can be difficult in numer- ical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Downward flows (or plumes) crossing a convective boundary at the bottom of an envelope are clearly observed in nu- merical simulations (see for example Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Ballistic plume crossings may eventually lead to a modification of the thermal background – the so-called penetration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' But for such modifi- cation to be observed, simulations must be run over many thousands of convective turnover timescales, as theoretically expected and re- cently demonstrated in simulations by Anders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2022) based on 3D simulations of convection in a Cartesian box with idealised se- tups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' In a numerical study of solar-like convective envelopes, Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2021) show that artificially boosting the luminosity of the stellar model by a factor 104 yields a significant modification of the thermal background below the convective boundary with an ex- tension of the size of the layer characterised by the penetration of convective flows, which could lead to a growth of the convectively unstable zone down to deeper levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Whether this growth stabilises or whether the convective boundary continues moving downward indefinitely is unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' For the solar-like model with realistic stellar luminosity, a slight modification of the thermal background is also observed in the simulations of Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2021), but they show no trend of an extension of the Schwarzschild convective boundary over the simulation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Following the approach developed in Pratt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2017) for con- vective envelopes, the most vigorous plumes can be used to define a maximal overshooting length, which can be significantly deeper than the typical length reached by the bulk of the plumes (Pratt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Vlaykov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Whether this bal- listic process is also observed for convective cores and can drive significant mixing is an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Arguments based on the dy- namics of convective motions and plumes suggest that mixing below a convective zone (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='g envelope overshooting) and above (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='g core overshooting) may indeed be different (Andrássy & Spruit 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Simple arguments based on the kinetic energy of a plume with typ- ical velocity and the restoring buoyancy force suggest very small overshooting lengths for the cores of low and intermediate mass zero-age-main-sequence (ZAMS) stars (Higl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' But these estimates are based on typical velocities without considering possi- ble extreme plume events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The situation could also be different for convective cores on the ZAMS and on the main-sequence respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Indeed, the building of a molecular weight gradient at the core boundary due to hydrogen burning in the core can hamper the lifting of heavier material by ballistic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' An entrainment process slowly eroding the convective boundary may thus dominate at some point over the ballistic process during the main sequence evolution, or both processes may coexist and contribute to mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' These ques- tions are still unsettled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Existing numerical simulations of convective cores have mostly focussed on one single stellar mass model, rather than a range of stellar masses (Meakin & Arnett 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Gilet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Edelmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Horst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Higl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Additionally, many of these works enhance the stellar luminosity of the model, to provide numerical stability, or to accelerate the thermal relaxation or the Mach number of the con- vective flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This artefact may artificially favour one process over the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' At this time, it is difficult to draw any firm conclusion re- garding the main mechanisms that drive CBM in stars and how their efficiency is affected with stellar mass and with the stage of evolution on the main sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' In this work devoted to convective cores, we study the efficiency for convective plumes to penetrate into the stable region as a function of stellar mass for ZAMS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' In the following we will refer to overshooting to describe this process, since we essentially describe the ballistic process and even if a modification of the temperature gradient is observed for the most luminous models (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 5), likely leading to penetration as defined by Zahn (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We perform two-dimensional (2D) numerical simulations of convective cores of ZAMS stellar models covering a range of stellar masses between 3 𝑀⊙ and 20 𝑀⊙ (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Our goal is to apply the framework of ex- treme plume events developed for convective stellar envelopes (Pratt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2017, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2021) to the convective cores of inter- mediate and massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We analyse whether extreme events can provide overshooting lengths required for stellar models to reproduce observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' For this purpose, we derive a relationship between over- shooting length and stellar luminosity based on present numerical simulations (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We apply the relationship to one-dimensional stellar evolution models and test them against observations (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This is the first step for a systematic study devoted to convective core overshooting in intermediate mass and massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2 NUMERICAL SIMULATIONS We use the fully compressible time-implicit code MUSIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' A full description of MUSIC and of the time-implicit integration can be found in Viallet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2011, 2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Goffrey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' MUSIC MNRAS 000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 1–12 (2022) A study of convective core overshooting as a function of stellar mass 3 solves the inviscid Euler equations in the presence of external gravity and thermal diffusion: 𝜕𝜌 𝜕𝑡 = −∇ · (𝜌v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (1) 𝜕𝜌v 𝜕𝑡 = −∇ · (𝜌v ⊗ v) − ∇𝑝 + 𝜌g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2) 𝜕𝜌𝑒 𝜕𝑡 = −∇ · (𝜌𝑒v) − 𝑝∇ · v + ∇ · (𝜒∇𝑇) + 𝑄nuc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (3) where 𝜌 is the density,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 𝑒 the specific internal energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' v the velocity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 𝑝 the gas pressure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 𝑇 the temperature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' g the gravitational accelera- tion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' and 𝜒 the thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The term 𝑄nuc represents the nuclear energy rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The symbol ⊗ is the outer product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' All hydrody- namical simulations presented in this work are performed assuming spherically symmetric gravitational acceleration g, which is updated every time interval Δ𝑡1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' All simulations presented in this work are performed with Δ𝑡 = 103 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The typical dynamical timescale of the entire stellar cores analysed in this study 𝜏dyn ∼ 1/ √︁ (𝜌mean𝐺), with 𝜌mean the mean density of the core and 𝐺 the gravitational constant, is of the order of 103 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We have checked with a number of test simulations that a variation of Δ𝑡 between 102 and 105 seconds does not impact our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' In the stellar models considered, radiative transfer is the major heat transport that contributes to the thermal conductivity, which is given for photons by 𝜒 = 16𝜎𝑇3 3𝜅𝜌 , (4) where 𝜅 is the Rosseland mean opacity, and 𝜎 the Stefan-Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Realistic stellar opacities and equation of states appropriate for the description of stellar interiors are implemented in MUSIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Opacities are interpolated from the OPAL tables (Iglesias & Rogers 1996) for solar metallicity and the equation of state is based on the OPAL EOS tables of Rogers & Nayfonov (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='1 Initial stellar models To provide the initial structures for the 2D simulations, we compute stellar models in the mass range 3-20 𝑀⊙ with the one-dimensional Lyon stellar evolution code (Baraffe & El Eid 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 1998), using the same opacities and equation of state as MUSIC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The 2D simulations require as initial input a radial profile of den- sity and internal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The 1D stellar evolution models have an initial helium abundance in mass fraction 𝑌=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='28 and solar metal- licity 𝑍=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='02 and were computed through the pre-main sequence and main sequence phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' All initial models for the 2D simulations in this study are taken at the beginning of core hydrogen burning and have a central abundance of helium 𝑌c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2838, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' only ∼ 1% of their central hydrogen has been depleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' There is thus a very shallow mean molecular weight gradient at the convective boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Follow-up analysis of later stages of evolution with a steeper gradient of molecular weight at the core boundary are in progress (Morison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' in prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Convective stability is defined by the Schwarzschild 1 Note that Δ𝑡 is the time after which the gravitational potential is updated, not the numerical timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The numerical timestep used for these simulations is set by the hydrodynamical CFL number varying between 10 and 50 (see Viallet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2011, for definitions) and corresponding to values for the timestep ranging between 5 s and 40 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2 The 1D initial structures are available on the repository http://perso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='ens- lyon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='fr/isabelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='baraffe/2Dcore_overshooting_2023 criterion ∇ < ∇ad, with ∇ = d log𝑇 d log 𝑃 the temperature gradient and ∇ad = d log𝑇 d log 𝑃 |𝑆 the adiabatic gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The 1D stellar models used to generate the initial structures for the 2D simulations do not account for overshooting at the convective core boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' In the following, we define the Schwarzschild boundary as the transition layer be- tween convective instability (∇ > ∇ad) and stability (∇ < ∇ad).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The properties of the initial 1D stellar structures are provided in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Nuclear energy generated in the convective cores is accounted for in the internal energy equation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (3)) through the term 𝑄nuc using the radial profile of the nuclear energy rate from the 1D stellar model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Given that the simulation times are orders of magnitude smaller than the nuclear timescale for H burning in the cores, the nuclear energy is assumed to remain constant with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2 Spherical-shell geometry and boundary conditions Two-dimensional simulations are performed in a spherical shell using spherical coordinates, namely 𝑟 the radius and 𝜃 the polar angle, and assuming azimuthal symmetry in the 𝜙-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' For all models, the inner radius 𝑟in is defined at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='02 𝑅star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The choice of the outer radius 𝑟out depends on the stellar model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Since the main motivation of this work is to analyse the extent of the overshooting layer for different stellar masses, the outer radius 𝑟out is fixed at a distance of ∼ 1 × 𝐻𝑃,CB for the lowest mass (3 𝑀⊙) to ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='5 × 𝐻𝑃,CB for the highest mass (20 𝑀⊙) away from the convective boundary 𝑟conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Extension of the radial domain to analyse the generation of internal waves at the core boundary and their propagation in the radiative envelope is work in progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The angular extent ranges from 𝜃 = 0◦ to 𝜃 = 180◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The grid has uniform spacing in the r and 𝜃 coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The choice for the resolution (𝑁𝑟, 𝑁𝜃) is set by the condition to have a good resolution of the pressure scale height at the Schwarzschild boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Effective Reynolds and Prandtl numbers are commonly used to set the resolution of numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' But given that our simulations are based on an implicit Large Eddy Simulation (ILES) approach, only a rough estimate can be provided for these numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' They will in any case remain far away from the conditions prevailing in stellar interiors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We suggest that a more relevant resolution criterion for hydrodynamical simulations devoted to the study of overshooting using realistic stellar structures should be the number of grid cells per pressure scale height at the convective boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This should allow a more relevant comparison between the works of different groups devoted to the study of different stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We use ∼ 110 − 140 grid cells per pressure scale height in the radial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The details of the resolution adopted in this work are provided in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We have also performed a few tests with higher resolution and analyse the impact in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The radial boundary conditions for the density correspond to a constant radial derivative on the density (see Pratt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The energy flux at the inner and outer radial boundaries are set to the value of the energy flux at that radius in the one-dimensional stellar evolution model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' At the boundaries in 𝜃, because of the extension of the angular domain to the poles, reflective boundary conditions for the density and energy are used (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' the values are mirrored at the boundary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' For the velocity, we impose reflective conditions at the radial and polar boundaries, corresponding to: v𝑟 = 0 and 𝜕v𝜃 𝜕𝑟 = 0 at 𝑟in and 𝑟out, 𝜕v𝑟 𝜕𝜃 = 0 and v𝜃 = 0 at 𝜃 = 0◦ and 𝜃 = 180◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We have also performed simulations for the 3 𝑀⊙ model with artificial enhancement of the stellar luminosity and the thermal dif- fusivity by factors 10, 102, 103 and 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This covers the range of MNRAS 000, 1–12 (2022) 4 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Properties of the initial stellar models (all models have a central helium abundance 𝑌c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2838) used for the 2D hydrodynamical simulations: total mass, stellar luminosity, stellar radius, mass and radius of the convective core (corresponding to the location of the Schwarzschild boundary) and the pressure scale height at the Schwarzschild boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 𝑀/𝑀⊙ 𝐿star/𝐿𝑎 ⊙ 𝑅star (cm) 𝑀conv/𝑀⊙ 𝑟conv/𝑅star 𝐻𝑃,CB (cm) 3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='7673 × 101 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='3855 × 1011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='5724 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='1486 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='3 × 1010 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2186 × 102 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='8424 × 1011 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='212 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='1814 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='8 × 1010 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='5726 × 103 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='7295 × 1011 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='046 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2239 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='7 × 1010 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='9242 × 104 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='4255 × 1011 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2580 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='3 × 1010 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2962 × 104 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='0172 × 1011 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='7947 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2869 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='7 × 1010 𝑎 We use 𝐿⊙ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='839 × 1033 erg/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Evolution of the total kinetic energy (in erg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' y-axis with a base-10 log scale) as a function of time (in s) for the simulations described in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Top panel: results for 3 𝑀⊙ models with various luminosity enhancement factors: 3L0 (black), 3L1 (blue), 3L2 (magenta), 3L3 (cyan) and 3L4 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Bottom panel: results for a range of stellar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The dotted line for each model corresponds to the value of the total kinetic energy at the beginning of the steady state for convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' luminosities of the stellar masses considered in this work (3-20 𝑀⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This choice of enhancement factor allows a comparative analysis of the impact of the luminosity for fixed core mass and increasing core mass, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Note that even larger enhancement factors (up to 107) for a 3 𝑀⊙ stellar structure can be found in previous works (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Edelmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' For the artificially boosted simulations, the energy flux (equivalently the luminosity) at the ra- dial boundaries is multiplied by the enhancement factor, the nuclear energy rate is multiplied by the same factor and the Rosseland mean opacities 𝜅 in MUSIC are decreased by the same factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Radial profile of the time averaged rms velocity (solid lines) and rms radial velocity (dashed lines) scaled by (𝐿star/1035)1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Top panel: re- sults for 3 𝑀⊙ models with various luminosity enhancement factors: 3L0 (black), 3L1 (blue), 3L2 (magenta), 3L3 (cyan) and 3L4 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Bottom panel: results for a range of stellar masses: 3L0 (black), 5L0 (blue), 10L0 (ma- genta),15L0 (red) and 20L0 (cyan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The convective boundary corresponding to the Schwarzschild boundary from the 1D initial model is indicated by a vertical solid line with the colour corresponding to each stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 3 RESULTS: AVERAGE DYNAMICS The properties of all simulations are summarised in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We de- fine 𝑡steady as the time required to reach a steady state for convection, characterised by the total kinetic energy 𝐸kin of the system reaching a plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Before 𝑡steady, the initial relaxation phase is characterised by the propagation of strong acoustic waves and the onset of convec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' At 𝑡steady, the value of the kinetic energy starts to stabilise and MNRAS 000, 1–12 (2022) A study of convective core overshooting as a function of stellar mass 5 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Main properties of the 2D simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Model 𝑀/𝑀⊙ 𝐿 (erg/s) 𝑁𝑟 × 𝑁𝜃 𝑟out/𝑅star 𝜏𝑎conv (s) 𝑁 𝑏 conv 𝑡𝑐 steady (s) 𝑡𝑑 sim (s) 3L0 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='981 ×1035 336 x 168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='9 ×106 1442 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='5 ×108 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='71 ×109 3L1 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='981 ×1036 336 x 168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='25 8 ×105 1211 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='6 ×108 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='43 ×109 3L2 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='981 ×1037 336 x 168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='9 ×105 501 9 ×107 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='84 ×108 3L2xhres 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='981 ×1037 684 x 342 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='8 ×105 514 9 ×107 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='84 ×108 3L3 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='981 ×1038 336 x 168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='7 ×105 1904 6 ×107 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='81×108 3L3xhres 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='981 ×1038 684 x 342 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='7 ×105 1243 6 ×107 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='71 ×108 3L4 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='981 ×1039 336 x 168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='25 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='9 ×104 1457 3 ×107 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='60 ×108 3L4xhres 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='981 ×1039 684 x 342 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='25 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='7 ×104 1400 3 ×107 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='52 ×108 5L0 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='003 ×1036 400 x 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='4 ×106 1260 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='45 ×108 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='01 ×109 10L0 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='139 ×1037 416 x 208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2 ×106 1260 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='1 × 108 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='77 ×109 15L0 15 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='387 ×1037 688 x 344 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='1 ×106 875 108 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='14×109 20L0 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='649 ×1038 864 x 430 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='1 ×106 800 9 × 107 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='99 ×108 𝑎 Convective turnover time (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 3 for its definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 𝑏 Number of convective turnover times covered by the simulation once steady state convection is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 𝑐Physical time to reach a steady state for convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 𝑑Total physical runtime of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' from this time it remains roughly constant with time (following the dotted curve which corresponds to the value of 𝐸kin at 𝑡steady for each model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The simulations are stopped at time 𝑡sim provided in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' None of these simulations are thermally relaxed, given that the total simulation times for all models are orders of magnitude smaller than the relevant thermal timescale ∼ 𝐺𝑀2/(𝑅star𝐿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' As a consequence all these simulations are expected to maintain a secular drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We have compared the radial profile of the internal energy, averaged in the angular direction, for each 2D model at time 𝑡steady and at time 𝑡sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We find a maximum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='5% relative difference for the internal energy at a given radius, with the largest difference found for the most luminous models (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The above-mentioned drift is thus so slow that calculating statistical or averaged data during this very slowly changing transitional state is sensible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Figure 1 shows the evolution of the total kinetic energy as a func- tion of time for all models and the plateau characterising their steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The initial transient phase can last a relatively long time, de- pending on the model studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' For the model 3L0, we note a dif- ferent behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' After the peak due to strong acoustic waves, the kinetic energy continuously decreases until 𝑡 ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='4 × 108 s (log 𝑡 ∼ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' In this regime, convection develops in the core (within the 1D Schwarzschild boundary) in two spatially separate regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The abrupt increase of 𝐸kin observed at 𝑡 ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='4×108 s marks the merging of these two convective regions and the beginning of fully developed convection in the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The Mach number characterising the con- vective velocities in model 3L0 is small, of the order of ∼ 10−4, which is numerically challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This low Mach number explains why several previous works artificially enhance the luminosity of the model (Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Horst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' There is no need for this artefact for the model 3L0 as MUSIC’s numerical scheme allows convection to develop and eventually reach a steady state even after a long transient phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Note that this unusual transient phase observed for the model 3L0 will likely change with a different procedure for initialising the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' All simulations start without an imposed background noise (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' initial velocities are set to zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Imposing initially a background noise for the model 3L0 may change the loca- tion where convection starts and thus the behaviour of the transient phase, which is irrelevant for the analysis performed in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' A global convective turnover time 𝜏conv is estimated based on the rms velocity vrms(𝑟, 𝑡) at radius 𝑟 and time 𝑡, which characterises a bulk convective velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We define 𝜏conv by: 𝜏conv = �∫ 𝑟conv 𝑟in d𝑟 vrms(𝑟, 𝑡) � 𝑡, (5) where the rms velocity is given by vrms(𝑟, 𝑡) = √︃ ⟨v2(𝑟, 𝜃, 𝑡)⟩𝜃, (6) with v2 = v2𝑟 + v2 𝜃, v𝑟 and v𝜃 being the radial and angular velocities, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Time averages are denoted by ⟨⟩𝑡 and calculated between 𝑡steady and 𝑡sim, the final time reached by the simulation (see values in Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' For any quantity 𝑋 we define: � 𝑋 � 𝑡 = 1 (𝑡sim − 𝑡steady) ∫ 𝑡sim 𝑡steady 𝑋d𝑡 (7) The volume-weighted average in the angular direction ⟨⟩𝜃 is defined for any quantity X as: � 𝑋(𝑟, 𝜃, 𝑡) � 𝜃 = ∫ 𝜃 𝑋(𝑟, 𝜃, 𝑡)d𝑉(𝑟, 𝜃) ∫ 𝜃 d𝑉(𝑟, 𝜃) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (8) The simulations are stopped after a time 𝑡sim when convergence of the statistics used to determine the size of the layer penetrated by plumes is obtained, as explained in the next section (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Table 2 provides the values and numbers of the convective turnover times, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Figure 2 displays the rms velocity and rms radial velocity for the 3 𝑀⊙ models with artificially enhanced luminosities (upper panel) and for the range of stellar masses investigated (lower panel), scaled by 𝐿1/3 star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' In the convective core, our simulations re- produce the expected scaling of convective velocity with luminosity vconv ∝ 𝐿1/3 recovered by many hydrodynamical simulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Edelmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Andrassy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Horst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Higl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This scaling is expected from mixing-length theory based on the argument that the turbulent dissipation rate of kinetic energy in a turbulent convective zone scales with v3 (Biermann 1932).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' But a general scaling of the total flux with v3 can also be derived for the kinetic energy and the enthalpy fluxes based on simple dimensional arguments (see Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2017) MNRAS 000, 1–12 (2022) 6 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The rms velocities in the stably stratified region are due to the penetrative flows just above the convective boundary and to the prop- agation of internal waves excited by the convective motions and the penetrating plumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2 shows that these ve- locities also increase with the luminosity, suggesting more efficient overshooting of the convective motions above the convective bound- ary and thus larger overshooting length with increasing luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2021) reports similar behaviours for convective en- velopes of solar-like models with artificially enhanced luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Quantitative estimate of the overshooting lengths for all models is performed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 4 RESULTS: EXTENT OF THE OVERSHOOTING REGION 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='1 Determination of overshooting lengths To determine an overshooting length, we adopt the same approach as in Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2021) and initially inspired by the findings of Pratt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This approach is based on the analysis of the depth of all convective plumes that penetrate beyond the convective boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The two criteria used to determine the depth of a penetrative plume at a given angle 𝜃 and time 𝑡 are based on the first zero above the convective boundary 𝑟conv of the vertical kinetic energy flux fk and vertical heat flux f𝛿T, defined by (see Pratt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2017): fk(𝑟, 𝜃, 𝑡) = 1 2 𝜌(𝑟, 𝜃, 𝑡)v2(𝑟, 𝜃, 𝑡)v𝑟 (𝑟, 𝜃, 𝑡), (9) f𝛿T(𝑟, 𝜃, 𝑡) = 𝜌(𝑟, 𝜃, 𝑡)𝑐𝑃(𝑟, 𝜃, 𝑡)𝛿𝑇(𝑟, 𝜃, 𝑡)v𝑟 (𝑟, 𝜃, 𝑡), (10) where 𝑐𝑃 is the specific heat at constant pressure and the temperature fluctuation 𝛿𝑇 is defined by: 𝛿𝑇(𝑟, 𝜃, 𝑡) = 𝑇(𝑟, 𝜃, 𝑡) − �� 𝑇(𝑟, 𝜃, 𝑡) � 𝜃 � 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (11) The method is the same as the one developed in Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2021) for convective envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' At each time 𝑡, we calculate at each angle 𝜃 the radial positions 𝑟0(𝜃, 𝑡) of a plume corresponding to the first zero of fk and f𝛿T, respectively, above the convective boundary 𝑟conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The corresponding overshooting length 𝑙0 with respect to 𝑟conv is defined by 𝑙0(𝜃, 𝑡) = 𝑟0(𝜃, 𝑡) − 𝑟conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (12) Figure 3 illustrates the angular structure of the overshooting layer at an arbitrary time for the 10 𝑀⊙ stellar model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We then define the maximal overshooting length 𝑙max 0 at a given time by the maximum over all angles 𝜃: 𝑙max 0 (𝑡) = max(𝑙0(𝜃, 𝑡)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (13) The time average 𝑙max = ⟨𝑙max 0 (𝑡)⟩𝑡 provides an effective width for the overshooting layer where the most vigorous plumes penetrate and which we use to characterise the extension of the mixing layer over the long term evolution of the star (Pratt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Table 3 displays 𝑙max based on the criterion for fk and f𝛿T, respectively, for all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The distributions of overshooting lengths derived from fk and f𝛿T, respectively, slowly converges with time, as found in Pratt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2017) and Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Several hundreds to thousand convective turnover times, depending on the stellar model, are required for the statistics to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Eventually, both criteria provide similar values for the effective overshooting width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The values of the overshooting width based on f𝛿T converge faster with time, compared to the value based on fk, as found as well for convective envelopes in Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The values of 𝑙max(f𝛿T) provided in Table 3 have reached a steady state for all Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Overshooting lengths𝑙0 defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (12) as a function of the angle 𝜃 at time 𝑡 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='3108s for the 10 𝑀⊙ model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The upper panel corresponds to 𝑙0 defined by fk and the lower panel to 𝑙0 defined by f𝛿T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The horizontal dashed line in each panel indicates the average overshooting length at this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' models after 𝑡sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Depending on the stellar model, 𝑙max(fk) gets close to 𝑙max(f𝛿T) (difference of <∼ 20%) for all models but models 3L0 and 20L0, for which 𝑙max(fk) continues slowly decreasing even after more than 800 ×𝜏conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We run three simulations for the 3 𝑀⊙ models with enhanced luminosity with twice the resolution in both radial and angular directions and covering about the same simulation time as their lower resolution counterpart, in order to check the sensitivity of the values of 𝑙max to the resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The properties of these higher resolution models (labelled 2xhres) are displayed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The results for the overshooting lengths are given in Table 3 and show similar values for lmax(f𝛿T) as found with a lower resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The values for 𝑙max(fk) of the higher resolution models are larger than the corresponding value for the lower resolution model, as it takes more time for 𝑙max(fk) in the high resolution models to decrease to the level of 𝑙max(f𝛿T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' But the value of 𝑙max(fk) in the high resolution models continues decreasing with time and we expect it to eventually converge and thus get much closer to 𝑙max(f𝛿T) and to the value of 𝑙max(fk) found in the lower resolution model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2 Relationship between overshooting length and stellar luminosity The variation of 𝑙max with the stellar luminosity is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 4 for the 3𝑀⊙ models with enhanced luminosity and for the set of stellar masses with realistic luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' As expected from the behaviour of the rms velocities (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2) overshooting lengths increase with the stellar luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' To derive an approximate scaling relationship for the overshooting length 𝑑ov that can be implemented in stellar evolution codes, we use the values of 𝑙max derived from f𝛿T, since these values have converged with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We derive the following MNRAS 000, 1–12 (2022) A study of convective core overshooting as a function of stellar mass 7 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Effective width 𝑙max of the overshooting layer in units of the total stellar radius and of the pressure scale height at the convective boundary, for all models considered in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The quantity 𝑙max(fk) is based on the criterion using fk (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 9) and 𝑙max(f𝛿T) is based on f𝛿T (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Model 𝑙max(fk)/𝑅star 𝑙max(f𝛿T)/𝑅star 𝑙max(fk)/𝐻𝑃,CB 𝑙max(f𝛿T)/𝐻𝑃,CB 3L0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='4 ×10−3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='7 ×10−3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='8 ×10−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='9 ×10−2 3L1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2 ×10−3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2 ×10−3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='5 × 10−2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='5 ×10−2 3L2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2 ×10−3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='1 ×10−3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='6 × 10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='5 ×10−2 3L2xhres 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='4 ×10−3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='4 ×10−3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='9 × 10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='8 ×10−2 3L3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='8 ×10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='6 ×10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='9 ×10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='7 ×10−1 3L3xhres 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2 ×10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='6 ×10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='3 ×10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='7 ×10−1 3L4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='5 ×10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='8 ×10−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='7 ×10−1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='0 ×10−1 3L4xhres 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='0 ×10−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='0 ×10−2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2 ×10−1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2×10−1 5L0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='3 ×10−3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='0 ×10−3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='5 ×10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='1 ×10−2 10L0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2 ×10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='1 ×10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2 ×10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='1 ×10−1 15L0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='6 ×10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='3 ×10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='66 ×10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='35 ×10−1 20L0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='5 ×10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='0 ×10−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='8 ×10−1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='17 ×10−1 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Overshooting length 𝑙max, in units of the pressure scale height at the convective boundary, as a function of the model luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The 3 𝑀⊙ models with various luminosity enhancement factors are indicated in red (dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The results for a range of stellar masses with realistic stellar luminosity are indicated in blue (solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The dotted curve shows the fit for the overshooting length 𝑑𝑜𝑣/𝐻P,CB given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' expression which fits the results for the stellar mass range studied: 𝑑ov/𝐻P,CB = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='05 × 10−3 × (𝐿/𝐿⊙)1/3 × (𝑟conv/𝐻𝑃,CB)1/2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='02 (14) We find a typical scaling with the luminosity 𝑑ov ∝ 𝐿1/3 ∝ vconv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Numerical studies of convective envelopes report overshooting lengths 𝑑ov which vary with the luminosity following 𝑑ov ∝ 𝐿𝑎 with 𝑎 varying between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='08 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='31 (Hotta 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Käpylä 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The analytical model of Zahn (1991) for pene- tration, based on first order estimate of the deceleration of a plume in an adiabatically stratified penetration zone, predicts 𝑑ov ∝ v3/2 conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Our results also show that the overshooting lengths derived for a fixed stellar mass (and thus a fixed convective core size) are systematically smaller than the one derived for larger cores but similar luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Interestingly, a dependence of 𝑑ov with the size of the core 𝑟conv is also predicted by Zahn (1991) (see their Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='5)) with the same re- lation of proportionality 𝑑ov ∝ (𝑟conv/𝐻𝑃,CB)1/2 as found in present simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This dependence in the Zahn model is derived from the strong variations with radius of various relevant quantities such as the gravitational acceleration 𝑔, the mass 𝑚(𝑟) enclosed in a sphere of radius 𝑟, the radiative conductivity 𝜒, and thus the radiative flux, close to the convective core boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' In our simulations, we expect the radial dependence of the gravitational acceleration to have the main impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We find that the larger the core (in terms of radius and mass), the smaller the gravitational acceleration at the core boundary 𝑔conv ∼ 𝐺𝑀conv/𝑟2conv (see values in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Therefore, the larger the stellar mass, the larger the velocities at the convective boundary and the smaller the restoring force due to gravity, implying up-flows to penetrate over larger distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This is a plausible explanation for the dependence of 𝑑ov on the convective core radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We analyse below (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 6) whether the expression provided by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (14) pro- vides a reasonable agreement between stellar evolution models and observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 5 THERMAL BACKGROUND EVOLUTION The prescription used in the previous section to determine overshoot- ing lengths relies on two assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Firstly, we consider that the simulations have reached a steady state for convection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' a global dynamical steady state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This assumption is reasonable based on the observation that the total kinetic energy of the system reaches a plateau as a function of time (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Secondly, we assume that the relevant convective boundary from which the overshooting lengths are defined is the 1D Schwarzschild boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This is directly useful for the purpose of implementing these overshooting lengths in 1D stellar evolution codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' However, we find that in all models a small nearly adiabatic layer just above the convective boundary forms rapidly once convection steady state is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' For the most luminous models, we observe that this small layer slowly grows in size with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Anders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2022) also find a modification of the temperature gradient which becomes close to the adiabatic gradient in the pen- etration layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' They report that their simulations exhibit the process of convective penetration as defined by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Zahn (1991), with con- vective penetrating motions mixing entropy and establishing a nearly adiabatic stratification above the Schwarzschild boundary (see also Brummell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Anders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2022) suggest that the extent of convective penetration is limited and derive arguments involving the MNRAS 000, 1–12 (2022) 8 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Visualisation of the radial velocity v𝑟 [cm/s] (top panel) and the relative temperature fluctuations (𝑇 −⟨𝑇 ⟩𝜃)/⟨𝑇 ⟩𝜃 (bottom panel) in a region zoomed around the convective boundary (horizontal black line) for the model 20L0 at time 𝑡 = 7 × 108 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The x-axis represents the co-latitude (in terms of cos 𝜃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Note that to the better illustrate upwellings and downwellings in the top panel, the velocity scale is saturated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' any velocity > vr,max = 5 × 103 cm/s (< vr,min = −5×103 cm/s) are represented with the same color as vr,max (vr,min).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' convective flux, the viscous dissipation rate and the buoyancy work, providing an estimate of the penetration width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Depending on their setup, they find that penetration zones can take thousands of con- vective turnover times to saturate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' They show properties of the flow and of the temperature fluctuations close to a convective boundary (see their Figure 1) which are similar to our results, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 5 for the model 20L0 at a given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' As expected in con- vective regions, convective upflows transport hot material from the central regions up to the top of the convective core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Inspection of temperature fluctuations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' the difference between the local tem- perature and the horizontally averaged thermal background) indeed indicates that upflows in the convective region are characterised by positive temperature fluctuations and downflows by negative tem- perature fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' When upflows cross the convective boundary, at the top of the convective core, and penetrate the stably stratified medium, they adiabatically expand and therefore get cooler (neg- ative temperature fluctuation) and denser than the subadiabatically stratified environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' To understand the establishment of a nearly adiabatic layer in the penetration region, one needs to compare the advection timescale, which characterises the process of entropy mixing by penetrating flows (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' an advection process), and the thermal diffusion timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' If penetrating flows, as illustrated in the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 5, can drive efficient entropy/thermal mixing, the layer characterised by pene- trating up-flows will remain nearly adiabatic if thermal diffusion is slow enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Table 4 provides estimates of the diffusive timescale Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Profile of the time and angular averages of the quantity (∇ − ∇ad) in the layers just above the convective core for the most luminous models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The 1D profile of (∇ − ∇ad) is indicated by the black dashed line and the 1D convective core boundary by the vertical solid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The location of 𝑙max derived from f𝛿T is indicated by the vertical dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' In both panels, the solid blue line corresponds to the time average between 𝑡steady and 𝑡sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The curves in magenta correspond to time averages over 20×𝜏conv at a given time, as indicated in each panel (time 𝑡 in s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 𝜏diff = 𝐿2/𝜅rad at the core boundary, with 𝐿 a relevant lengthscale and 𝜅rad = 𝜒/(𝜌𝑐𝑃) the thermal diffusivity (which is the radiative diffusivity for present stellar models with 𝜒 defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Esti- mate of an advection timescale 𝜏adv = 𝐿/vr,rms is based on the time averaged rms radial velocity at the core boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' For the charac- teristic lengthscales at the core boundary, we use the overshooting distance 𝑙max(f𝛿T) (see Table 3) and the pressure scale height 𝐻P (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' As illustrated in Table 4, typical advection timescales are much smaller than typical thermal diffusion timescales for all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The growth in size with time of the nearly adiabatic layer observed in the most luminous models is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 6 for the models 3L3 and 3L4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This growth with time may also happen in the less luminous models, but their very slow evolution and less vigorous penetrating flows may prevent clearly exhibiting this feature over present simulation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We also note that the angular averaged temperature gradient in the models, while getting very close to the adiabatic gradient, remains stable against the Schwarzschild criterion over the simulation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' For the purpose of analysing the time evolution of the nearly adiabatic layer, we have extended the simulation time of the models 3L3 and 3L4 beyond the value of 𝑡sim used to determine overshooting depths (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2), until 𝑡final = 5 × 108 s (∼ 2600 × 𝜏conv for 3L3 and ∼ 5300×𝜏conv for 3L4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The aim is to reach a simulation time for these models close to or greater than the thermal diffusion timescale in the overshooting layer 𝜏diff(𝑙max).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Given the smaller grid size and larger thermal diffusivity of these models, this is still computationally affordable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Figure 6 shows clearly in models 3L3 and 3L4 that the radial extension of the nearly adiabatic layer slows down with time MNRAS 000, 1–12 (2022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='36 4000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='32 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='30 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='26 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='24 4000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='00 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='75 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='50 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='00 cos θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='36 10-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='34 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='32 10-5 10-6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='30 T- (T)e)/ f−, we find that bands 1 and 2 have non-zero Chern +numbers +1 and -1 (Fig. 3f). Under the opposite condition, +f+ < f−, the Chern numbers reverse sign as seen in Fig. 3g. +When f+ = f−, TRS is preserved, and all bands have Chern + +5 +a +c +d +b +𝒇! = 𝟎. 𝟑 +𝒇" = 𝟎 +𝒇! = 𝟎. 𝟑 +𝒇" = 𝟎 +S3 +Band 1 +Band 2 +𝒇! = 𝟎 +𝒇" = 𝟎. 𝟑 +𝒇! = 𝟎 +𝒇" = 𝟎. 𝟑 +Band 1 +Band 2 +S3 +S3 +S3 +FIG. 4. The Stokes parameter, S3(k), which is a measure of the +degree of circular polarization of a mode (Eq. 8), under pumping +with (a,c) σ+ polarized light which creates populations f+ = 0.3, +f− = 0 and (b,d) σ− polarized light which creates populations f+ = +0, f− = 0.3 of the two lowest energy bands (Band 1 and 2 as indicated +in the inset). We used parameters β0 = 0.1eV, β = 9×10−4eVµm2, +ζ = 2.5 × 10−3eVµm, m∗ = 125¯h2eV−1µm−2, E0 = 3.80eV and +¯hωe = 3.81eV (see Supplementary section S4 for details). +number 0 as seen in Fig. 3e and 3h. In Fig. 3b-c, we plot the +computed Berry curvature when f+ ̸= f− and due to broken +TRS, we find Ωl(k) ̸= −Ωl(−k). +We also plot the Stokes parameter, S3(k), for bands 1 and +2, under pumping with circularly polarized light, in Fig. 4. +The Stokes parameter, S3(k), provides information on the de- +gree of circular polarization of the photonic component of an +exciton-polariton band and is calculated as +S3(k) = |b+,cav(k)|2 −|b−,cav(k)|2 +|b+,cav(k)|2 +|b−,cav(k)|2 +(8) +where +the +eigenvectors +of +the +band +are +��ul,k +� += +b+,cav(k)|+cav⟩ + b−,cav(k)|−cav⟩ + b+,mol(k)|+mol⟩ + +b−,mol(k)|−mol⟩. In the absence of pumping, we find that +within a band, one half of the modes are predominantly +σ+ polarized and the other half are σ− polarized (Fig. +3e). Once TRS is broken with circularly polarized optical +pumping, a large number of modes within each band become +overwhelmingly of the same polarization (Fig. 3f-g and Fig. +4). +In experiments, the Berry curvature of photon bands in a +Fabry-Perot cavity can be extracted from the components of +the Stokes vector [25, 26]. However, in the case of exciton- +polariton bands, the Berry curvature of only sections of the +band that are predominantly photonic and have negligible +molecular character [49] can be measured experimentally as, +to the best of our knowledge, it is difficult to obtain the phase +relationship between the photonic and molecular components, +unless light-matter cross-correlation functions are measured. +Therefore, in our case, the Berry curvature of only parts of the +band that are mostly photonic in Fig. 3a-d can be measured +using pump-probe spectroscopy. This measurement should +be feasible as long as the time delay between the pump and +probe pulses is shorter than the time the system takes to depo- +larize and reach a state with f+ = f−. As the depolarization +timescale for porphyrins ranges from 210 fs to 1.6 ps, this +measurement should be viable [50]. +As the Chern numbers of bands 1 and 2 are modified +through pumping with circularly polarized light, if we per- +form a calculation where a region of the system is pumped +with σ+ polarized light ( f+ ̸= 0 and f− = 0) and an adjacent +region is pumped with σ− polarized light (f+ = 0 and f− ̸= 0), +we expect edge states at the boundary between these regions. +However, as our Hamiltonian does not contain couplings be- +tween neighboring molecules, and the position of a molecule +does not enter the Hamiltonian anywhere except through the +phase of the light-matter coupling eik·rm, the standard bulk- +boundary correspondence is no longer applicable and we do +not observe edge states. We do not include plots for these cal- +culations in this work and leave it an open question whether +there is an analogous statement for bulk-boundary correspon- +dence in these types of systems. +On the other hand, for +exciton-polariton systems where nearest-neighbor couplings +are present, edge states have been predicted and observed +[19, 20]. +Other systems +To emphasize that our scheme of saturating electronic tran- +sitions with circularly polarized light to modify topological +properties is not limited to organic exciton-polariton systems, +we compute the Berry curvature of two other polariton sys- +tems where porphyrin is replaced with (i) Ce:YAG and (ii) +MoS2 (Fig. 5a and 5d). Other materials can also be used in +place of porphyrins, as long as they have transitions that can +be selectively excited with circularly polarized light and these +transitions have large enough transition dipole moments that +they can couple strongly to the photon modes of a cavity. +In Yttrium Aluminum garnet (YAG) doped with Cerium, +Ce3+ ions replace some Y3+ and Ce3+ has transitions that can +be selectively excited with circularly polarized light. Here, +each Ce3+ has two possible ground states, one with the elec- +tron in spin up |4 f(1) ↑⟩, and the other with it in spin down +|4 f(1) ↓⟩. Similarly, it has a degenerate pair of excited spin +states |5d(1) ↑⟩ and |5d(1) ↓⟩. +The |4 f(1) ↓⟩ ↔ |5d(1) ↑⟩ +transition has ∼ 400 times larger oscillator strength for ex- +citation with σ+ polarized light than with σ− polarized light, +therefore, we take the transition dipole moment to be µµµ+ (Fig. +5b) [51]. Similarly, we take the transition dipole to be µµµ− for +the |4f(1) ↑⟩ ↔ |5d(1) ↓⟩ transition (Fig. 5b). The transitions +in Ce:YAG do couple to photon modes, however, to the best +of our knowledge, strong coupling has not been reported in +the literature [52, 53]. Nevertheless, strong light-matter cou- +pling has been achieved with a similar system: Nd3+ doped +YSO and YVO crystals [54, 55], and based on our calcula- +tions, with a 0.1µm thick sample of Ce:YAG at concentration +1% Ce3+ (relative to Y3+), we should be able to attain strong + +6 +Ce:YAG +a +b +d +f +e +c +|4f(1)↓⟩ +|5d(1)↑⟩ +|5d(1)↓⟩ +𝝁! +𝝁" +|4f(1)↑⟩ +𝑓↓ = 0.4 +𝑓↑ = 0.6 +𝑓# = 0.3 +𝑓#! = 0 +𝝁" +𝝁! +K +K’ +Ω1 (𝜇m2) +Ω1 (𝜇m2) +FIG. 5. (a) Illustration of Ce:YAG (salmon block) and perylene (green blocks) within a Fabry-Perot cavity. (b) Atomic levels of Ce3+ +ions embedded in Yttrium Aluminum garnet (YAG) where the yellow circles indicate the fraction f↓ of Ce3+ ions in the |4f(1) ↓⟩ state +and the fraction f↑ in the |4f(1) ↑⟩ state after optical pumping. The transition dipoles µµµ± = µ0(ˆx ± iˆy)/ +√ +2 are also indicated. (c) Berry +curvature of the lowest energy band, Ω1(k), under pumping with circularly polarized which creates populations f↓ = 0.4 and f↑ = 0.6. (d) +Illustration of monolayer MoS2 and perylene (green blocks) within a Fabry-Perot cavity. (e) Illustration of A-excitons in the K and K’ valleys +of monolayer MoS2. (f) Berry curvature of the lowest energy band, Ω1(k), under pumping with circularly polarized which creates exciton +populations fK = 0.3 and fK′ = 0. We used parameters β0 = 0.1eV, β = 9×10−4eVµm2, ζ = 2.5×10−3eVµm, m∗ = 125¯h2eV−1µm−2, (c) +E0 = 2.50eV, ¯hωe = 2.53eV and (f) E0 = 1.80eV, ¯hωe = 1.855eV (see Supplementary section S4 for details). +coupling with photon modes in a Fabry-Perot cavity (see Sup- +plementary section S4). +Under thermal equilibrium, the populations of the |4f(1) ↑⟩ +and |4 f(1) ↓⟩ states are equal. However, under pumping with +pulses of σ+ polarization, in the presence of a small magnetic +field ∼ 0.049T, the population of |4 f(1) ↑⟩ will exceed that +of |4 f(1) ↓⟩ because population is selectively removed from +|4f(1) ↓⟩ and added to |5d(1) ↑⟩ by the circularly polarized +pulses, but decay from the excited |5d(1) ↑⟩ state to the two +ground states has equal probability [56]. In principle, a mag- +netic field is not required; however, as we do not know the spin +relaxation time in the absence of the magnetic field, we report +the magnetic field used in the experimental study [56]. Under +optical pumping with circularly polarized light, the 5d states +will have very small populations which we take to be zero, +while the |4f(1) ↓⟩ and |4 f(1) ↑⟩ states will have unequal +populations f↓ and f↑, respectively; here, f↓ + f↑ = 1. Op- +tically pumped Ce:YAG can then be modeled using the effec- +tive Hamiltonian in eq. 6 and 7, with |±mol⟩′ → |5d(1) ↑ / ↓⟩ +and √1− f∓ −2 f± → � f↓/↑. The large spin relaxation time +of ∼ 4.5 ms makes this system particularly well-suited for +our scheme because it maintains f↓ ̸= f↑, and hence non-zero +Chern invariants, for an extended period of time [56]. In Fig. +5c we plot Berry curvature of the lowest band of a perylene +filled cavity strongly coupled with Ce:YAG, where f↓ = 0.4 +and f↑ = 0.6 (see Supplementary section S4 for values of other +parameters). +TMDs, such as single-layer MoS2, display optically con- +trollable valley polarization and could also be used in place +of porphyrins [57–59]. +Due to lack of inversion symme- +try in these systems, the K and K’ valleys are inequivalent; +this results in optical selection rules that allow selective cre- +ation of excitons at K and K’ valleys with σ+ and σ− polar- +ized light, respectively [60, 61]. Additionally, strong light- +matter coupling has been observed when monolayer MoS2 is +placed within a Fabry-Perot cavity [8, 9]. This system has +depolarization times of ∼ 200fs - 5ps making it possible to +measure Berry curvature using pump-probe spectroscopy be- +fore depolarization occurs [62, 63]. We model this exciton- +polariton system (Fig. 5d) using eq. 6 and eq. 7 (we focus +on the A-exciton, see Supplementary section S4 for parame- +ters) with |+mol⟩ → |K⟩, |−mol⟩ → |K′⟩ and √1− f∓ −2 f± → +�1−2 fK/K′. In Fig. 5f we plot the Berry curvature of the +lowest band when fK = 0.3 and fK′ = 0. Unfortunately, sig- +nificant Rabi contraction upon optical pumping has not been +experimentally observed in these systems which will make it +challenging to observe Berry curvature as in Fig. 5f since +our model relies on saturation effects. However, for exciton +polaritons formed from monolayer TMDs, even if Rabi con- +traction through resonant optical pumping may not produce +the intended effect, off-resonant optical pumping can break +the degeneracy of excitons in the K and K’ valleys through +optical stark effect [64], and this may have interesting conse- +quences for the Berry curvature. Additionally, if bilayer MoS2 +is used in place of monolayer MoS2, effects on the Berry cur- +vature described in our work may be more pronounced as bi- +layer MoS2 hosts interlayer excitons which possess large op- +tical nonlinearities; specifically, they display saturation and + +7 +Rabi contraction under strong coupling [65, 66]. +Finally, so far we have only considered replacing porphyrin +with a different material, such as MoS2 or Ce:YAG. In addi- +tion to this, perylene can also be replaced with other suitable +materials. In our work, we choose to use a cavity filled with +perylene because we do not want degeneracy at any k within +the photon bands. Other systems also satisfy this requirement +and could be used instead. For instance, we could use an elec- +trically tunable, highly anisotropic, liquid-crystal cavity with +well separated H and V polarized photon modes [24, 67]. A +perovskite cavity is another potential candidate due to its high +anisotropy, and optical pumping may help lift the degeneracy +of polariton modes in this system [49]. Additionally, other +photonic structures can also be used instead of a cavity, as +long as the photon bands are not degenerate at any k and have +non-zero light-matter coupling at all k. +CONCLUSION +In summary, we show that TRS can be broken in organic +exciton-polariton systems through selectively saturating elec- +tronic transitions with a circularly polarized pump and that the +resulting bands possess non-zero Chern invariants. In particu- +lar, we demonstrate this theoretically for a Fabry-Perot cavity +filled with porphyrin and perylene. The Berry curvature of +the more photonic parts of the bands of this system can be +measured experimentally using pump-probe spectroscopy, as +long as the time delay is shorter than the depolarization time +for porphyrin (210fs-1.6ps) [50], and this will reveal non-zero +Berry curvature and Chern number under circularly polarized +pumping. Our scheme relies on Rabi contraction from satu- +ration of optical transitions. It is important to note that edge +states do not emerge in our system despite non-zero Chern in- +variants as our model does not contain sufficient positional +information about the molecules or the unit cells. Bleu et +al. [30] have previously proposed breaking TRS in inorganic +exciton-polariton systems through pumping with circularly +polarized light, however, their work relies on polariton con- +densation and having patterned lattices. Finally, we demon- +strate that saturating electronic transitions to modify topol- +ogy is not limited to organic systems. To illustrate this, we +calculate the Berry curvature and Chern numbers of exciton- +polariton bands of two other systems under optical pumping: +(a) Ce:YAG and (b) monolayer MoS2, and find similar results +as the organic exciton-polariton case. In view of recent devel- +opments on electrically tuning the Berry curvature of liquid- +crystal and perovskite filled cavities [24, 26], our work pro- +vides an additional control knob to optically tune the Berry +curvature of exciton-polariton systems using circularly polar- +ized light. Additionally, ultrafast control of topological prop- +erties of systems with light may find use in nonreciprocal and +nonlinear optoelectronic devices. +ACKNOWLEDGEMENTS +S.P.-S. acknowledges support from NSF Grant No. CA- +REER CHE 1654732 for the development of the model and +calculations. +The conceptualization of the molecular and +solid-state systems was guided by N.P.S. and J.Y.-Z. as part of +the Center for Molecular Quantum Transduction (CMQT), an +Energy Frontier Research Center funded by the U.S. Depart- +ment of Energy, Office of Science, Basic Energy Sciences un- +der Award No. DE-SC0021314. S.P.-S. thanks Kai Schwen- +nicke and Stephan van den Wildenberg for useful discussions. +CODE AVAILABILITY +Code available at https://github.com/SindhanaPS/Topological_Polaritons_Submission. +REFERENCES +[1] Claude Weisbuch, Mr Nishioka, A Ishikawa, and Y Arakawa. +Observation of the coupled exciton-photon mode splitting in a +semiconductor quantum microcavity. Physical Review Letters, +69(23):3314, 1992. +[2] R André, D Heger, Le Si Dang, and Y Merle d’Aubigné. Spec- +troscopy of polaritons in cdte-based microcavities. Journal of +crystal growth, 184:758–762, 1998. +[3] David G Lidzey, DDC Bradley, MS Skolnick, T Virgili, +S Walker, and DM Whittaker. Strong exciton–photon coupling +in an organic semiconductor microcavity. Nature, 395(6697): +53–55, 1998. +[4] R Butté, G Christmann, E Feltin, J-F Carlin, M Mosca, +M Ilegems, and N Grandjean. Room-temperature polariton lu- +minescence from a bulk gan microcavity. Physical Review B, +73(3):033315, 2006. +[5] R Shimada, J Xie, Vitaliy Avrutin, Ü Özgür, and H Morkoˇc. +Cavity polaritons in zno-based hybrid microcavities. Applied +Physics Letters, 92(1):011127, 2008. +[6] Antoine Brehier, Radoslav Parashkov, Jean-Sébastien Lauret, +and Emmanuelle Deleporte. Strong exciton-photon coupling +in a microcavity containing layered perovskite semiconductors. +Applied physics letters, 89(17):171110, 2006. +[7] Rui Su, Antonio Fieramosca, Qing Zhang, Hai Son Nguyen, +Emmanuelle Deleporte, Zhanghai Chen, Daniele Sanvitto, Tim- +othy CH Liew, and Qihua Xiong. Perovskite semiconductors +for room-temperature exciton-polaritonics. Nature Materials, +20(10):1315–1324, 2021. +[8] Xiaoze Liu, Tal Galfsky, Zheng Sun, Fengnian Xia, Erh- +chen Lin, Yi-Hsien Lee, Stéphane Kéna-Cohen, and Vinod M +Menon. +Strong light–matter coupling in two-dimensional +atomic crystals. Nature Photonics, 9(1):30–34, 2015. +[9] Fengrui Hu and Zhe Fei. Recent progress on exciton polaritons +in layered transition-metal dichalcogenides. Advanced Optical +Materials, 8(5):1901003, 2020. +[10] James A Hutchison, Tal Schwartz, Cyriaque Genet, Eloïse De- +vaux, and Thomas W Ebbesen. Modifying chemical landscapes +by coupling to vacuum fields. +Angewandte Chemie Interna- +tional Edition, 51(7):1592–1596, 2012. +[11] KS Daskalakis, SA Maier, Ray Murray, and Stéphane Kéna- +Cohen. Nonlinear interactions in an organic polariton conden- +sate. Nature materials, 13(3):271–278, 2014. + +8 +[12] Christof P Dietrich, Anja Steude, Laura Tropf, Marcel Schu- +bert, Nils M Kronenberg, Kai Ostermann, Sven Höfling, and +Malte C Gather. An exciton-polariton laser based on biolog- +ically produced fluorescent protein. +Science advances, 2(8): +e1600666, 2016. +[13] Nina Krainova, Alex J Grede, Demetra Tsokkou, Natalie +Banerji, and Noel C Giebink. Polaron photoconductivity in the +weak and strong light-matter coupling regime. Physical review +letters, 124(17):177401, 2020. +[14] Qing Liao, Charly Leblanc, Jiahuan Ren, Feng Li, Yiming Li, +Dmitry Solnyshkov, Guillaume Malpuech, Jiannian Yao, and +Hongbing Fu. +Experimental measurement of the divergent +quantum metric of an exceptional point. Physical Review Let- +ters, 127(10):107402, 2021. +[15] Marco Dusel, Simon Betzold, Tristan H Harder, Monika Em- +merling, Johannes Beierlein, Jürgen Ohmer, Utz Fischer, Ronny +Thomale, Christian Schneider, Sven Hofling, et al. +Room- +temperature topological polariton laser in an organic lattice. +Nano Letters, 21(15):6398–6405, 2021. +[16] Dmitry D Solnyshkov, Guillaume Malpuech, Philippe St-Jean, +Sylvain Ravets, Jacqueline Bloch, and Alberto Amo. Micro- +cavity polaritons for topological photonics. Optical Materials +Express, 11(4):1119–1142, 2021. +[17] Charles-Edouard Bardyn, Torsten Karzig, Gil Refael, and Tim- +othy CH Liew. Topological polaritons and excitons in garden- +variety systems. Physical Review B, 91(16):161413, 2015. +[18] Joel Yuen-Zhou, Semion K Saikin, Tony Zhu, Mehmet C On- +basli, Caroline A Ross, Vladimir Bulovic, and Marc A Baldo. +Plexciton dirac points and topological modes. Nature commu- +nications, 7(1):1–7, 2016. +[19] S Klembt, TH Harder, OA Egorov, K Winkler, R Ge, MA Ban- +dres, M Emmerling, L Worschech, TCH Liew, M Segev, et al. +Exciton-polariton topological insulator. +Nature, 562(7728): +552–556, 2018. +[20] Torsten Karzig, Charles-Edouard Bardyn, Netanel H Lindner, +and Gil Refael. Topological polaritons. Physical Review X, 5 +(3):031001, 2015. +[21] Wenjing Liu, Zhurun Ji, Yuhui Wang, Gaurav Modi, Minsoo +Hwang, Biyuan Zheng, Volker J Sorger, Anlian Pan, and Ritesh +Agarwal. Generation of helical topological exciton-polaritons. +Science, 370(6516):600–604, 2020. +[22] Mengyao Li, Ivan Sinev, Fedor Benimetskiy, Tatyana Ivanova, +Ekaterina Khestanova, Svetlana Kiriushechkina, Anton Vaku- +lenko, Sriram Guddala, Maurice Skolnick, Vinod M Menon, +et al. +Experimental observation of topological z2 exciton- +polaritons in transition metal dichalcogenide monolayers. Na- +ture communications, 12(1):1–10, 2021. +[23] A Gianfrate, O Bleu, L Dominici, V Ardizzone, M De Giorgi, +D Ballarini, G Lerario, KW West, LN Pfeiffer, DD Solnyshkov, +et al. Measurement of the quantum geometric tensor and of the +anomalous hall drift. Nature, 578(7795):381–385, 2020. +[24] Katarzyna Rechci´nska, Mateusz Król, Rafał Mazur, Prze- +mysław Morawiak, Rafał Mirek, Karolina Łempicka, Witold +Bardyszewski, Michał Matuszewski, Przemysław Kula, Wik- +tor Piecek, et al. Engineering spin-orbit synthetic hamiltonians +in liquid-crystal optical cavities. Science, 366(6466):727–730, +2019. +[25] Jiahuan Ren, Qing Liao, Feng Li, Yiming Li, Olivier Bleu, +Guillaume Malpuech, Jiannian Yao, Hongbing Fu, and Dmitry +Solnyshkov. Nontrivial band geometry in an optically active +system. Nature communications, 12(1):1–8, 2021. +[26] Karolina Łempicka-Mirek, Mateusz Król, Helgi Sigurdsson, +Adam Wincukiewicz, Przemysław Morawiak, Rafał Mazur, +Marcin Muszy´nski, Wiktor Piecek, Przemysław Kula, Tomasz +Stefaniuk, et al. Electrically tunable berry curvature and strong +light-matter coupling in liquid crystal microcavities with 2d +perovskite. Science Advances, 8(40):eabq7533, 2022. +[27] Sriram Guddala, Yuma Kawaguchi, Filipp Komissarenko, Svet- +lana Kiriushechkina, Anton Vakulenko, Kai Chen, Andrea Alù, +Vinod M Menon, and Alexander B Khanikaev. +All-optical +nonreciprocity due to valley polarization pumping in transi- +tion metal dichalcogenides. Nature communications, 12(1):1–9, +2021. +[28] Erik J Lenferink, Guohua Wei, and Nathaniel P Stern. Coher- +ent optical non-reciprocity in axisymmetric resonators. Optics +express, 22(13):16099–16111, 2014. +[29] Kai Schwennicke and Joel Yuen-Zhou. Optical activity from +the exciton aharonov–bohm effect: A floquet engineering ap- +proach. The Journal of Physical Chemistry C, 124(7):4206– +4214, 2020. +[30] O Bleu, DD Solnyshkov, and Guillaume Malpuech. Photonic +versus electronic quantum anomalous hall effect. Physical Re- +view B, 95(11):115415, 2017. +[31] Teng Long, Xuekai Ma, Jiahuan Ren, Feng Li, Qing Liao, +Stefan Schumacher, Guillaume Malpuech, Dmitry Solnyshkov, +and Hongbing Fu. Helical polariton lasing from topological val- +leys in an organic crystalline microcavity. Advanced Science, 9 +(29):2203588, 2022. +[32] Mercedes Rubio, Björn O Roos, Luis Serrano-Andrés, and +Manuela Merchán. Theoretical study of the electronic spectrum +of magnesium-porphyrin. The Journal of chemical physics, 110 +(15):7202–7209, 1999. +[33] Lawrence Edwards, David H Dolphin, and Martin Gouterman. +Porphyrins: Xvi. vapor absorption spectra and redox reactions: +Octalkylporphins. Journal of Molecular Spectroscopy, 35(1): +90–109, 1970. +[34] Tonatiuh Rangel, Andre Rinn, Sahar Sharifzadeh, Felipe H +da Jornada, André Pick, Steven G Louie, Gregor Witte, Leeor +Kronik, Jeffrey B Neaton, and Sangam Chatterjee. Low-lying +excited states in crystalline perylene. Proceedings of the Na- +tional Academy of Sciences, 115(2):284–289, 2018. +[35] Ingo Barth, Jörn Manz, Yasuteru Shigeta, and Kiyoshi Yagi. +Unidirectional electronic ring current driven by a few cycle cir- +cularly polarized laser pulse: quantum model simulations for +mg- porphyrin. Journal of the American Chemical Society, 128 +(21):7043–7049, 2006. +[36] Joel Yuen-Zhou, Semion K Saikin, Norman Y Yao, and Alán +Aspuru-Guzik. Topologically protected excitons in porphyrin +thin films. Nature materials, 13(11):1026–1032, 2014. +[37] Shichao Sun, Bing Gu, and Shaul Mukamel. Polariton ring cur- +rents and circular dichroism of mg-porphyrin in a chiral cavity. +Chemical science, 13(4):1037–1048, 2022. +[38] Shaul Mukamel. Principles of nonlinear optical spectroscopy. +Number 6. Oxford University Press on Demand, 1999. +[39] Vladimir M Agranovich. Excitations in organic solids, volume +142. OUP Oxford, 2009. +[40] Alexey V Kavokin, Jeremy J Baumberg, Guillaume Malpuech, +and Fabrice P Laussy. Microcavities, volume 21. Oxford uni- +versity press, 2017. +[41] Giovanna Panzarini, Lucio Claudio Andreani, A Armitage, +D Baxter, MS Skolnick, VN Astratov, JS Roberts, Alexey V +Kavokin, Maria R Vladimirova, and MA Kaliteevski. Exciton- +light coupling in single and coupled semiconductor microcav- +ities: Polariton dispersion and polarization splitting. Physical +Review B, 59(7):5082, 1999. +[42] Mário G Silveirinha. Chern invariants for continuous media. +Physical Review B, 92(12):125153, 2015. + +9 +[43] Uzi Even, Jacob Magen, Joshua Jortner, Joel Friedman, +and Haim Levanon. +Isolated ultracold porphyrins in super- +sonic expansions. i. free-base tetraphenylporphyrin and zn- +tetraphenylporphyrin. The Journal of Chemical Physics, 77(9): +4374–4383, 1982. +[44] S Voelker, RM Macfarlane, AZ Genack, HP Trommsdorff, and +JH van Der Waals. Homogeneous linewidth of the s 1← s 0 +transition of free-base porphyrin in an n-octane crystal as stud- +ied by photochemical hole-burning. The Journal of Chemical +Physics, 67(4):1759–1765, 1977. +[45] Bo Xiang, Raphael F Ribeiro, Adam D Dunkelberger, Jiaxi +Wang, Yingmin Li, Blake S Simpkins, Jeffrey C Owrutsky, Joel +Yuen-Zhou, and Wei Xiong. Two-dimensional infrared spec- +troscopy of vibrational polaritons. Proceedings of the National +Academy of Sciences, 115(19):4845–4850, 2018. +[46] Timur Yagafarov, Denis Sannikov, Anton Zasedatelev, Kyriacos +Georgiou, Anton Baranikov, Oleksandr Kyriienko, Ivan She- +lykh, Lizhi Gai, Zhen Shen, David Lidzey, et al. Mechanisms +of blueshifts in organic polariton condensates. Communications +Physics, 3(1):1–10, 2020. +[47] Raphael F. Ribeiro, Adam D Dunkelberger, Bo Xiang, Wei +Xiong, Blake S Simpkins, Jeffrey C Owrutsky, and Joel Yuen- +Zhou. Theory for nonlinear spectroscopy of vibrational polari- +tons. +The journal of physical chemistry letters, 9(13):3766– +3771, 2018. +[48] Andrew H Salij, Randall H Goldsmith, and Roel Tempelaar. +Chiral polaritons based on achiral fabry-perot cavities using ap- +parent circular dichroism. +arXiv preprint arXiv:2208.14461, +2022. +[49] Laura Polimeno, Giovanni Lerario, Milena De Giorgi, Luisa +De Marco, Lorenzo Dominici, Francesco Todisco, Annalisa +Coriolano, Vincenzo Ardizzone, Marco Pugliese, Carmela T +Prontera, et al. Tuning of the berry curvature in 2d perovskite +polaritons. Nature nanotechnology, 16(12):1349–1354, 2021. +[50] C Galli, Klaas Wynne, Steven M LeCours, MJ Therien, and +RM Hochstrasser. Direct measurement of electronic dephasing +using anisotropy. Chemical physics letters, 206(5-6):493–499, +1993. +[51] Roman Kolesov, Kangwei Xia, Rolf Reuter, Mohammad Ja- +mali, Rainer Stöhr, Tugrul Inal, Petr Siyushev, and Jörg +Wrachtrup. Mapping spin coherence of a single rare-earth ion +in a crystal onto a single photon polarization state. Physical +review letters, 111(12):120502, 2013. +[52] Robert J Moerland, I Gerward C Weppelman, Marijke Scotuzzi, +and Jacob P Hoogenboom. Nanoscale imaging of light-matter +coupling inside metal-coated cavities with a pulsed electron +beam. Nano Letters, 18(10):6107–6112, 2018. +[53] SRK Rodriguez, S Murai, MA Verschuuren, and J Gómez Ri- +vas. Light-emitting waveguide-plasmon polaritons. Physical +review letters, 109(16):166803, 2012. +[54] Tian Zhong, Jonathan M Kindem, Evan Miyazono, and Andrei +Faraon. Nanophotonic coherent light–matter interfaces based +on rare-earth-doped crystals. Nature communications, 6(1):1– +6, 2015. +[55] Tian Zhong, Jonathan M Kindem, Jake Rochman, and An- +drei Faraon. Interfacing broadband photonic qubits to on-chip +cavity-protected rare-earth ensembles. Nature communications, +8(1):1–7, 2017. +[56] P Siyushev, K Xia, R Reuter, M Jamali, N Zhao, N Yang, +C Duan, N Kukharchyk, AD Wieck, R Kolesov, et al. Coherent +properties of single rare-earth spin qubits. Nature communica- +tions, 5(1):1–6, 2014. +[57] Kin Fai Mak, Keliang He, Jie Shan, and Tony F Heinz. Control +of valley polarization in monolayer mos2 by optical helicity. +Nature nanotechnology, 7(8):494–498, 2012. +[58] Hualing Zeng, Junfeng Dai, Wang Yao, Di Xiao, and Xiaodong +Cui. Valley polarization in mos2 monolayers by optical pump- +ing. Nature nanotechnology, 7(8):490–493, 2012. +[59] Aswini Kumar Pattanayak, Pritam Das, Devarshi Chakrabarty, +Avijit Dhara, Shreya Paul, Satyait Maji, Maruthi Manoj Brun- +davanam, and Sajal Dhara. Probing spin dynamics of 2d ex- +citons with twisted light. +ACS Photonics, 9(10):3351–3356, +2022. +[60] Liuyang Sun, Chun-Yuan Wang, Alex Krasnok, Junho Choi, +Jinwei Shi, Juan Sebastian Gomez-Diaz, André Zepeda, +Shangjr Gwo, Chih-Kang Shih, Andrea Alù, et al. Separation +of valley excitons in a mos2 monolayer using a subwavelength +asymmetric groove array. +Nature Photonics, 13(3):180–184, +2019. +[61] Guan-Hao Peng, Oscar Javier Gomez Sanchez, Wei-Hua Li, +Ping-Yuan Lo, and Shun-Jen Cheng. Twisted-light-induced ex- +citon wave packets in transition-metal dichalcogenide monolay- +ers. arXiv preprint arXiv:2203.02081, 2022. +[62] Stefano Dal Conte, Federico Bottegoni, Eva Arianna Aurelia +Pogna, D De Fazio, Stefano Ambrogio, Ilaria Bargigia, Cosimo +D’Andrea, A Lombardo, M Bruna, Franco Ciccacci, et al. Ul- +trafast valley relaxation dynamics in monolayer mos 2 probed +by nonequilibrium optical techniques. Physical Review B, 92 +(23):235425, 2015. +[63] Yen-Jung Chen, Jeffrey D Cain, Teodor K Stanev, Vinayak P +Dravid, and Nathaniel P Stern. +Valley-polarized exciton– +polaritons in a monolayer semiconductor. Nature Photonics, +11(7):431–435, 2017. +[64] Trevor LaMountain, Jovan Nelson, Erik J Lenferink, Samuel H +Amsterdam, Akshay A Murthy, Hongfei Zeng, Tobin J Marks, +Vinayak P Dravid, Mark C Hersam, and Nathaniel P Stern. +Valley-selective optical stark effect of exciton-polaritons in a +monolayer semiconductor. Nature communications, 12(1):1–7, +2021. +[65] Biswajit +Datta, +Mandeep +Khatoniar, +Prathmesh +Desh- +mukh, Félix Thouin, Rezlind Bushati, Simone De Liberato, +Stephane Kena Cohen, and Vinod M Menon. +Highly non- +linear dipolar exciton-polaritons in bilayer mos2. +Nature +communications, 13(1):1–7, 2022. +[66] Charalambos Louca, Armando Genco, Salvatore Chiavazzo, +Thomas P Lyons, Sam Randerson, Chiara Trovatello, Pe- +ter Claronino, Rahul Jayaprakash, Kenji Watanabe, Takashi +Taniguchi, et al. Nonlinear interactions of dipolar excitons and +polaritons in mos2 bilayers. arXiv preprint arXiv:2204.00485, +2022. +[67] Marcin Muszy´nski, Mateusz Król, Katarzyna Rechci´nska, +Przemysław Oliwa, Mateusz K˛edziora, Karolina Łempicka- +Mirek, Rafał Mazur, Przemysław Morawiak, Wiktor Piecek, +Przemysław Kula, et al. +Realizing persistent-spin-helix las- +ing in the regime of rashba-dresselhaus spin-orbit coupling in +a dye-filled liquid-crystal optical microcavity. Physical Review +Applied, 17(1):014041, 2022. + +Molecular and solid-state topological polaritons via optical saturation: supplemental document +Sindhana Pannir-Sivajothi,1 Nathaniel P. Stern,2 and Joel Yuen-Zhou1, ∗ +1Department of Chemistry and Biochemistry, University of California San Diego, La Jolla, California 92093, USA +2Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA +S1. +LIGHT-MATTER COUPLING +The light-matter coupling part of the total Hamiltonian under the electric dipole approximation is, +ˆHcav−mol =∑ +m ∑ +k,α +− ˆµµµm · ˆEk,α(rm,0), +=∑ +m ∑ +k,α +− +� +∑ +α′=± +(µµµα′ ˆσ† +m,α′ + µµµ∗ +α′ ˆσm,α′) +� +· ˆEk,α(rm,0), +(S1) +where µµµα′ = µµµm,α′ = +� +m,α′ +mol +�� ˆµµµ |m,G⟩ is independent of m since we assume that all porphyrin molecules lie flat in the x-y +plane and are oriented. The electric field operator of the mode labeled by k and α is +ˆEk,α(r,z) = +� +¯hωk,α +2Vεε0 +� +f∗ +k,α(r,z) ˆa† +k,α +fk,α(r,z) ˆak,α +� +. +(S2) +Here, V = LxLyLz is the volume of the box we consider, where as mentioned in the main manuscript, we apply periodic boundary +conditions along the x and y directions. From here on, we will call the in-plane area of the box A = LxLy. Here, fk,α(r,z) is the +mode profile and it satisfies[1] +� +dr +� Lz +0 +dzf∗ +k,α(r,z)fk,α(r,z) = LzA. +(S3) +For the TE and TM modes[2], +fk,TE(r,z) =eik·r√ +2sin +� +nzπ +Lz +� +z+ Lz +2 +�� +ˆφφφ, +fk,TM(r,z) =eik·r +� +2 +|k|2 + +� nzπ +Lz +�2 +��nzπ +Lz +� +sin +� +nzπ +Lz +� +z+ Lz +2 +�� +ˆρρρ −i|k|cos +� +nzπ +Lz +� +z+ Lz +2 +�� +ˆz +� +. +(S4) +We make the rotating-wave approximation, +ˆHcav−mol =∑ +m ∑ +k,α +− +� +∑ +α′=± +(µµµα′ ˆσ† +m,α′ + µµµ∗ +α′ ˆσm,α′) +� +· +�� +¯hωk,α +2Vεε0 +� +f∗ +k,α(rm,0) ˆa† +k,α +fk,α(rm,0) ˆak,α +�� +, +≈ ∑ +m,α′ ∑ +k,α +− +� +¯hωk,α +2Vεε0 +� +µµµα′ ·fk,α(rm,0) ˆσ† +m,α′ ˆak,α + µµµ∗ +α′ ·f∗ +k,α(rm,0) ˆσm,α′ ˆa† +k,α +� +, += ∑ +m,α′ ∑ +k,α +� eik·rm +�NxNy +(µµµα′ ·Jk,α) ˆσ† +m,α′ ˆak,α + e−ik·rm +�NxNy +(µµµ∗ +α′ ·J∗ +k,α) ˆσm,α′ ˆa† +k,α +� +, +(S5) +where Jk,α = −�NxNy +� +¯hωk,α +2Vεε0 e−ik·rfk,α(r,0) and µµµα′ ·Jk,α is the collective light-matter coupling strength. +The annihilation operators of photon modes polarized along the horizontal (H) or x-axis and vertical (V) or y-axis are ˆak,H +and ˆak,V, respectively. They are related to α = ± polarized modes through ˆak,± = +1 +√ +2( ˆak,H ∓i ˆak,V)[3]. In addition, we assume +∗ joelyuen@ucsd.edu +arXiv:2301.03287v1 [physics.chem-ph] 9 Jan 2023 + +2 +that they are related to the TM and TE modes through ˆak,TM = cosφ ˆak,H +sinφ ˆak,V and ˆak,TE = −sinφ ˆak,H +cosφ ˆak,V. Using +this, we obtain the relationship between ˆak,TE, ˆak,TM and ˆak,+, ˆak,− modes to be, +ˆak,TM = 1 +√ +2 +� +eiφ ˆak,+ +e−iφ ˆak,− +� +, +ˆak,TE = 1 +√ +2 +� +ieiφ ˆak,+ −ie−iφ ˆak,− +� +. +(S6) +It is important to note that, based on these relationships and S4, the α =H/V modes are not completely linearly polarized and +the α = ± modes are not completely circularly polarized when |k| becomes comparable with nzπ/Lz. We also find, +Jk,+ = eiφ +√ +2 +� +Jk,TM +iJk,TE +� +, +Jk,− =e−iφ +√ +2 +� +Jk,TM −iJk,TE +� +. +(S7) +To keep the collective coupling strength µµµα′ ·Jk,α constant while taking the a → 0 limit, we take the magnitude of the collective +transition dipole of the bright state �NxNyµ0 over square root of the quantization area of the photon mode +√ +A to be a constant; +that is, we keep √ρAµ0 = µ0/a a constant, where ρA = NxNy/A is the areal density of quantum emitters. +Jk,α =−√ρA +� +¯hωk,α +2Lzεε0 +e−ik.rfk,α(r,0) +=− 1 +a +� +¯hωk,α +2Lzεε0 +e−ik.rfk,α(r,0). +(S8) +S2. +CHERN NUMBER CALCULATION +a +b +(kmax,kmax) +(kmax,-kmax) +(-kmax,-kmax) +(-kmax,kmax) +x +y +FIG. S1. (a) This is a cartoon figure that demonstrates the way Berry flux and Chern number are computed in our system. The small squares +are the plaquettes over which Berry flux is computed. The blue arrows specify the orientation used for Berry flux computation. Note that the +direction is opposite for the small squares and the large square. (b) Same as (a), but placed on a sphere. Here, it is more clear that the direction +of the arrow for the large square indicates the way Berry flux is computed for the giant plaquette covering the rest of the sphere. +For the Chern invariant to be an integer, it is important that the Berry curvature is integrated over a closed and bounded +surface [4]. For periodic systems with a finite period, the Brillouin zone is a torus which satisfies this requirement. However, + +3 +for a continuous system, (kx,ky) lies on an unbounded plane; for such systems, Silveirinha[5] proposed mapping this infinitely +large plane onto a sphere to compute the Chern number. This is the procedure we follow in our work. We discretize k-space and +compute the Berry flux in each plaquette within a square-shaped region in k-space, −kmax ≤ kx,ky ≤ kmax [4, 6] (Fig. S1a and +S1b). The entire region that satisfies the condition kx,ky > kmax or kx,ky < −kmax is taken as a single giant plaquette (Fig. S1b), +and the Berry flux within this region is computed by taking the Berry phase along the boundary of the plaquette but in a direction +opposite to that used to compute Berry flux for plaquettes within the square −kmax ≤ kx,ky ≤ kmax as indicated in Fig. S1a and +S1b. To ensure that we obtain a converged Chern number, we calculate the Chern number for different kmax and find that, for +our system, once kmax ≳ 100µm−1, the Chern number converges to C1 = ±1,C2 = ∓1,C3 = 0, and C4 = 0 when f+ ̸= f− with +|f+ − f−| ≳ 0.11. Smaller differences between f+ and f−, |f+ − f−| ≲ 0.11 require larger kmax for convergence. This is not a +problem for the f+ = f− case because the Chern invariant will always be zero due to time-reversal symmetry Ωl(k) = −Ωl(−k), +and we can use kmax ≈ 100µm−1 to compute it. +S3. +OPTICAL PUMPING +The number of excitations in the system Nex = ∑k,α a† +k,αak,α + ∑n,α σ† +n,ασn,α is a conserved quantity of this Hamiltonian. +Therefore, when we have f+ fraction of molecules in the |+mol⟩ state and f− in the |−mol⟩ state, we will only have to look at +the ( f+ + f−)Nth excitation manifold. Unfortunately, the dimensions of the Hilbert space of this manifold scale as +� +N +(f++f−)N +� +, +and this quickly becomes computationally intractable as the system size, N, increases. Using mean-field theory, we reduce this +many-body problem to a one-body problem. That is, we derive an effective Hamiltonian for a single excitation in the mean-field +of the remaining ( f+ + f−)N excitations; in this way, we reduce the dimensions of the Hilbert space to that of the first excitation +manifold. To do this, we follow a procedure similar to that used by Ribeiro et al.[7] and write the Heisenberg equations of +motion (EOM) for the operators ˆσm,± and ˆak,±, +i¯hd ˆσn,± +dt += +� ˆσn,±, ˆHmol +� ++ +� ˆσn,±, ˆHcav +� ++ +� ˆσn,±, ˆHcav−mol +� +=¯hωe ˆσn,± + +1 +�NxNy ∑ +k +eik·rn +� +(1− ˆσ† +n,∓ ˆσn,∓ −2 ˆσ† +n,± ˆσn,±) +� +Jk,+ · µµµ± ˆak,+ ++Jk,− · µµµ± ˆak,− +� +− ˆσ† +n,∓ ˆσn,± +� +Jk,+ · µµµ∓ ˆak,+ +Jk,− · µµµ∓ ˆak,− +�� +, +i¯hd ˆak,± +dt += +� +ˆak,±, ˆHmol +� ++ +� +ˆak,±, ˆHcav +� ++ +� +ˆak,±, ˆHcav−mol +� += +� +E0 + ¯h2|k|2 +2m∗ ±ζ|k|cosφ +� +ˆak,± + +� +−β0 +β|k|2e∓i2φ� +ˆa∓,k ++ +1 +�NxNy ∑ +m +eik·rm +� +J∗ +k,± · µµµ∗ ++ ˆσm,+ +J∗ +k,± · µµµ∗ +− ˆσm,− +� +. +(S9) +We make a mean-field approximation to linearize these EOM. For instance, we use mn ≈ ¯mn, that is, +ˆσ† +n,+ ˆσn,+ ˆak,+ = +� +⟨ ˆσ† +n,+ ˆσn,+⟩+ ˆσ† +n,+ ˆσn,+ −⟨ ˆσ† +n,+ ˆσn,+⟩ +� +ˆak,+ +=⟨ ˆσ† +n,+ ˆσn,+⟩ ˆak,+ +( ˆσ† +n,+ ˆσn,+ −⟨ ˆσ† +n,+ ˆσn,+⟩)⟨ ˆak,+⟩ +≈⟨ ˆσ† +n,+ ˆσn,+⟩ ˆak,+, +(S10) +where ⟨ ˆO⟩ = Tr +� ˆρ0 ˆO +� +with ˆρ0 ≈ ∏m ˆρm ∏k ∏α=+,− ˆρα,k[8]. Here, we assume that after dephasing of the molecular amplitudes, +ˆρm = fG |m,G⟩⟨m,G|+ f+ |m,+mol⟩⟨m,+mol|+ f− |m,−mol⟩⟨m,−mol|, ˆρα,k = |k,αcav,0⟩⟨k,αcav,0|, and, therefore, ⟨ ˆak,+⟩ = +0. The EOM then become +i¯hd ˆσn,± +dt +≈¯hωe ˆσn,± + +1 +�NxNy +(1− f∓ −2 f±)∑ +k +eik·rn +� +Jk,+ · µµµ± ˆak,+ ++Jk,− · µµµ± ˆak,− +� +, +i¯hd ˆak,± +dt += +� +E0 + ¯h2|k|2 +2m∗ ±ζ|k|cosφ +� +ˆak,± + +� +−β0 +β|k|2e∓i2φ� +ˆa∓,k ++ +1 +�NxNy ∑ +m +eik·rm +� +J∗ +k,± · µµµ∗ ++ ˆσm,+ +J∗ +k,± · µµµ∗ +− ˆσm,− +� +. +(S11) + +4 +We define rescaled operators ˆσ′ +n,± = ˆσn,±/√1− f∓ −2 f± and rewrite the EOM, +i¯hd ˆσ′ +n,± +dt +≈¯hωe ˆσ′ +n,± + +1 +�NxNy +� +1− f∓ −2f±∑ +k +eik·rn +� +Jk,+ · µµµ± ˆak,+ ++Jk,− · µµµ± ˆak,− +� +, +i¯hd ˆak,± +dt += +� +E0 + ¯h2|k|2 +2m∗ ±ζ|k|cosφ +� +ˆak,± + +� +−β0 +β|k|2e∓i2φ� +ˆa∓,k ++ �NxNy ∑ +m +eik·rm +�� +1− f− −2f+J∗ +k,± · µµµ∗ ++ ˆσ′ +m,+ + +� +1− f+ −2 f−J∗ +k,± · µµµ∗ +− ˆσ′ +m,− +� +. +(S12) +From these EOM, along with the fact that ˆσ′ +n,± act effectively as bosonic operators in mean-field, +� +ˆσ′ +n,+, ˆσ′† +n,+ +� += +1− ˆσ† +n,− ˆσn,−−2 ˆσ† +n,+ ˆσn,+ +1−f−−2 f+ +≈ +ˆI and +� +ˆσ′ +n,+, ˆσ′† +n,− +� += +− ˆσ† +n,− ˆσn,+ +1−f−−2 f+ ≈ ˆ0, where ˆI and ˆ0 are the identity and zero operators, we can construct an effective Hamiltonian +ˆHeff = ˆHeff +mol + ˆHeff +cav + ˆHeff +cav−mol in ˆσ′ +n,± and ˆak,±, +ˆHeff +mol =∑ +n +� +¯hωe ˆσ′† +n,+ ˆσ′ +n,+ + ¯hωe ˆσ′† +n,− ˆσ′ +n,− +� +, +ˆHeff +cav =∑ +k +� +E0 + ¯h2|k|2 +2m∗ +ζ|k|cosφ +� +ˆa† +k,+ ˆak,+ ++ +� +E0 + ¯h2|k|2 +2m∗ −ζ|k|cosφ +� +ˆa† +k,− ˆak,− + +� +−β0 +β|k|2e−i2φ� +ˆa† +k,+ ˆak,− ++ +� +−β0 +β|k|2ei2φ� +ˆa† +k,− ˆak,+, +ˆHeff +cav−mol = +1 +�NxNy ∑ +m ∑ +k +eik·rm +� +� +1− f− −2 f+ +� +Jk,+ · µµµ+ ˆσ′† +m,+ ˆak,+ ++Jk,− · µµµ+ ˆσ′† +m,+ ˆak,− +� ++ +� +1− f+ −2f− +� +Jk,+ · µµµ− ˆσ′† +m,− ˆak,+ ++Jk,− · µµµ− ˆσ′† +m,− ˆak,− +�� ++H.c., +(S13) +which is the mean-field Hamiltonian when the system has f+, f− excitations. Writing this effective Hamiltonian in k-space, +ˆHeff +mol =∑ +k +� +¯hωe ˆσ′† +k,+ ˆσ′ +k,+ + ¯hωe ˆσ′† +k,− ˆσ′ +k,− +� +, +ˆHeff +cav =∑ +k +� +E0 + ¯h2|k|2 +2m∗ +ζ|k|cosφ +� +ˆa† +k,+ ˆak,+ + +� +E0 + ¯h2|k|2 +2m∗ −ζ|k|cosφ +� +ˆa† +k,− ˆak,− ++ +� +−β0 +β|k|2e−i2φ� +ˆa† +k,+ ˆak,− + +� +−β0 +β|k|2ei2φ� +ˆa† +k,− ˆak,+, +ˆHeff +cav−mol =∑ +k +� +� +1− f− −2f+ +� +Jk,+ · µµµ+ ˆσ′† +k,+ ˆak,+ ++Jk,− · µµµ+ ˆσ′† +k,+ ˆak,− +� ++ +� +1− f+ −2f− +� +Jk,+ · µµµ− ˆσ′† +k,− ˆak,+ ++Jk,− · µµµ− ˆσ′† +k,− ˆak,− +�� ++H.c. +(S14) +We define states |k,±mol⟩′ and |k,±cav⟩′ corresponding to operators ˆσ′† +k,± and ˆa† +k,±, respectively. Writing the Hamiltonian +ˆHeff(k) = ⟨k| ˆHeff |k⟩ in the above basis we obtain, +ˆHeff(k) = ˆHeff +mol(k)+ ˆHeff +cav(k)+ ˆHeff +cav−mol(k), +(S15) + +5 +where, +ˆHeff +mol(k) =¯hωe |+mol⟩′ ⟨+mol|′ + ¯hωe |−mol⟩′ ⟨−mol|′ , +ˆHeff +cav(k) = +� +E0 + ¯h2|k|2 +2m∗ +ζ|k|cosφ +� +|+cav⟩′ ⟨+cav|′ + +� +E0 + ¯h2|k|2 +2m∗ −ζ|k|cosφ +� +|−cav⟩′ ⟨−cav|′ ++ +� +−β0 +β|k|2e−i2φ� +|+cav⟩′ ⟨−cav|′ + +� +−β0 +β|k|2ei2φ� +|−cav⟩′ ⟨+cav|′ , +ˆHeff +cav−mol(k) =Jk,+ · +�� +1− f− −2 f+µµµ+ |+mol⟩′ + +� +1− f+ −2 f−µµµ− |−mol⟩′ � +⟨+cav|′ ++Jk,− · +�� +1− f− −2 f+µµµ+ |+mol⟩′ + +� +1− f+ −2f−µµµ− |−mol⟩′ � +⟨−cav|′ +H.c. +(S16) +S4. +PARAMETERS +A. +Perylene filled cavity +We take parameters for the perylene filled cavity β0 = 0.1eV, β = 9×10−4eVµm2, ζ = 2.5×10−3eVµm, m∗ = 125¯h2eV−1µm−2, +and Lz = 0.745µm, where these are similar to those used to model the experiments of Ren et al.[9] (Fig. 3, 4, and 5 in main +manuscript). On the other hand, we modify E0 and nz such that they make the photon modes in our model near resonant with +the transition that is strongly coupled to the cavity. For instance, we take E0 = 3.80eV and nz = 11 for porphyrin (Fig. 3 and 4); +E0 = 2.50eV and nz = 9 for Ce:YAG (Fig. 5b-c); and E0 = 1.80eV and nz = 5 for MoS2 (Fig. 5e-f). We assume that perylene +has a similar effect on these different photon modes, as it does on modes with E0 ∼ 2.27eV at k = 0 in experiments[9]. This +may not necessarily be true, however, as we consider a perylene filled cavity only to achieve frequency separation of photon +modes with different polarization, and this can instead be easily achieved with an electrically tunable liquid crystal cavity [10], +replacing a perylene filled cavity with a liquid-crystal cavity will not modify the underlying physics of the phenomenon we are +interested in, i.e., the idea of using saturation to break TRS will remain intact. +B. +Porphyrin, Ce:YAG, and monolayer MoS2 +We take areal density ρA = 3.55 × 105µm−2 (∼ 2000 molecules in 75nm ××× 75nm)[11], relative permittivity ε = 1.5[12], +frequency ¯hωe = 3.8056eV and transition dipole µ0 = 1.1184au × 2.5417D/au = 2.84D [13] for the porphyrin film. Also, we +consider 100 such porphyrin films stacked one over the other along the z direction within the cavity to achieve strong light-matter +coupling, Nz = 100. Therefore, the effective areal density of molecules ρ′ +A = NzρA will be used instead of ρA while computing +Jk,α. These are the parameters used to generate Fig. 3 and 4. +Similarly, using density ρYAG = 5.11g cm−3, molar mass MYAG = 738 g mol−1, number of Y3+ per unit cell nY3+ = 3, and +concentration of Ce3+ (relative to Y3+) 1% = 10−2 [14], we obtain the effective areal density of Ce3+ ions in a L′ +z = 0.1µm +thick layer of Ce:YAG to be ρ′ +A = 10−2L′ +znY3+ρYAGNA/MYAG = 1.25 × 107µm−2. This will be used while computing Jk,α in +place of ρA. We use relative permittivity ε = 12[15] and frequency ¯hωe = 2.53eV (489nm[16]) for the transition in a Ce:YAG +crystal. Using the oscillator strength of this transition 0.286[16], we calculate the transition dipole µ0 = 5.46D. These are the +parameters used to generate Fig. 5c. +For monolayer MoS2, we consider A-excitons at ¯hωe = 1.855eV[17]. From Chen et al.[17], we take the Rabi splitting at +resonance, and use µ0√ρA +� +¯hωe/2Lzεε0 ≈ 39meV/2 = 19.5meV in our calculations (Fig. 5f). +SUPPLEMENTARY REFERENCES +[1] Fabre, C. & Treps, N. Modes and states in quantum optics. Reviews of Modern Physics 92, 035005 (2020). +[2] Zoubi, H. & La Rocca, G. Microscopic theory of anisotropic organic cavity exciton polaritons. Physical Review B 71, 235316 (2005). +[3] Martinelli, M. & Martelli, P. Polarization, mirrors, and reciprocity: birefringence and its compensation in optical retracing circuits. +Advances in Optics and Photonics 9, 129–168 (2017). +[4] Asb´oth, J. K., Oroszl´any, L. & P´alyi, A. A short course on topological insulators. Lecture notes in physics 919, 166 (2016). +[5] Silveirinha, M. G. Chern invariants for continuous media. Physical Review B 92, 125153 (2015). +[6] Fukui, T., Hatsugai, Y. & Suzuki, H. Chern numbers in discretized brillouin zone: efficient method of computing (spin) hall conductances. +Journal of the Physical Society of Japan 74, 1674–1677 (2005). +[7] F. Ribeiro, R. et al. Theory for nonlinear spectroscopy of vibrational polaritons. The journal of physical chemistry letters 9, 3766–3771 +(2018). + +6 +[8] Fowler-Wright, P., Lovett, B. W. & Keeling, J. Efficient many-body non-markovian dynamics of organic polaritons. Physical Review +Letters 129, 173001 (2022). +[9] Ren, J. et al. Nontrivial band geometry in an optically active system. Nature communications 12, 1–8 (2021). +[10] Rechci´nska, K. et al. Engineering spin-orbit synthetic hamiltonians in liquid-crystal optical cavities. Science 366, 727–730 (2019). +[11] Hulsken, B. et al. Real-time single-molecule imaging of oxidation catalysis at a liquid–solid interface. Nature nanotechnology 2, 285–289 +(2007). +[12] Li, D., Swanson, B. I., Robinson, J. M. & Hoffbauer, M. A. Porphyrin based self-assembled monolayer thin films: synthesis and +characterization. Journal of the American Chemical Society 115, 6975–6980 (1993). +[13] Sun, S., Gu, B. & Mukamel, S. Polariton ring currents and circular dichroism of mg-porphyrin in a chiral cavity. Chemical Science +(2022). +[14] Bachmann, V., Ronda, C. & Meijerink, A. Temperature quenching of yellow ce3+ luminescence in yag: Ce. Chemistry of Materials 21, +2077–2084 (2009). +[15] Ctibor, P., Sedl´aˇcek, J. & Hudec, T. Dielectric properties of ce-doped yag coatings produced by two techniques of plasma spraying. +Bolet´ın de la Sociedad Espa˜nola de Cer´amica y Vidrio (2021). +[16] Kolesov, R. et al. Mapping spin coherence of a single rare-earth ion in a crystal onto a single photon polarization state. Physical review +letters 111, 120502 (2013). +[17] Chen, Y.-J., Cain, J. D., Stanev, T. K., Dravid, V. P. & Stern, N. P. Valley-polarized exciton–polaritons in a monolayer semiconductor. +Nature Photonics 11, 431–435 (2017). + diff --git a/2tE1T4oBgHgl3EQflgTe/content/tmp_files/load_file.txt b/2tE1T4oBgHgl3EQflgTe/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..266c627a71333cb4380bbd2e30e505ada6282461 --- /dev/null +++ b/2tE1T4oBgHgl3EQflgTe/content/tmp_files/load_file.txt @@ -0,0 +1,959 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf,len=958 +page_content='Molecular and solid-state topological polaritons via optical saturation Sindhana Pannir-Sivajothi,1 Nathaniel P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Stern,2 and Joel Yuen-Zhou1, ∗ 1Department of Chemistry and Biochemistry, University of California San Diego, La Jolla, California 92093, USA 2Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA Strong coupling between electronic excitations in materials and photon modes results in the formation of hybrid quasiparticles called polaritons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Polariton systems often display larger nonlinearities than their photonic counterparts due to their material component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In this work, we theoretically investigate how to optically control the topological properties of molecular and solid-state exciton-polariton systems by exploiting one such nonlin- earity: saturation of electronic transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We study an optically pumped film of porphyrin molecules strongly coupled to the photon modes of a perylene filled Fabry-Perot cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Here, optical pumping with circularly polarized light breaks time-reversal symmetry instead of the frequently used large magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We can op- tically tune properties such as the Berry curvature and Chern numbers of the bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Importantly, while optical pumping does lead to non-zero Chern invariants, unidirectional edge states do not emerge in our system as the bulk-boundary correspondence is not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Finally, we illustrate the broad applicability of our scheme by computing the Berry curvature of two other systems with slightly modified level structures that lead to different nonlinear behavior when placed in a microcavity and pumped with circularly polarized light: (a) monolayer MoS2 and (b) Ce:YAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' This work demonstrates a versatile platform to control topological properties of hybrid light-matter systems to enrich the toolbox of optoelectronic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' INTRODUCTION Exciton-polaritons are hybrid excitations that exist in sys- tems where photonic modes couple strongly with optical tran- sitions in materials and their coupling strength exceeds losses [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Electronic strong coupling (ESC), where the optical tran- sitions correspond to semiconductor excitons or molecular electronic transitions, has been observed in a wide variety of inorganic and organic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' While some polariton sys- tems, such as GaAs and CdTe quantum wells in microcavi- ties [1, 2], often require cryogenic temperatures for operation, due to their small exciton binding energies, organic materials [3] along with others such as GaN [4], ZnO [5], perovskites [6, 7], and transition metal dichalcogenides (TMD) [8, 9] can FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Illustration of the system under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Porphyrin (molecules at the center) and perylene (green blocks) placed within a Fabry- Perot cavity and pumped with circularly polarized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ∗ joelyuen@ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='edu achieve ESC at room temperature when placed in Fabry-Perot cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In particular, organic exciton-polaritons have re- ceived attention for their ability to modify chemical reactiv- ity [10], demonstrate polariton condensation at room temper- ature [11, 12], improve photoconductivity [13], and display topological properties [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Exciton-polariton systems are versatile platforms for topo- logical applications as their hybrid nature provides the unique opportunity to take advantage of the nonlinearities and mag- netic response of the material component while still enjoy- ing benefits of the coherence properties of the photonic part [16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In the presence of photonic lattices, they also offer the possibility of unidirectional transport of energy through edge states that are robust to disorder [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' A few approaches are frequently used to achieve topological exciton-polariton bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In one of the approaches, the non-trivial topology re- sides in the winding light-matter coupling rather than individ- ual photon or exciton components [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' However, it is limited in application due to the requirement of large mag- netic fields to break time-reversal symmetry (TRS) and low temperatures to achieve Zeeman splitting in the exciton com- ponent which exceeds the exciton linewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In another ap- proach, TRS is preserved and a quantum spin hall insulator analogue is created in a polariton system [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' This approach does not require a large magnetic field, however, there, a topo- logical polariton system is created by coupling a topologically non-trivial photonic lattice with a topologically trivial exciton system and the interesting topology is almost entirely encoded in the photonic component of the polariton [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Both the approaches mentioned above were experimentally realized in polariton lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' More recently, polaritons in Fabry-Perot cavities have emerged as a viable platform for topological po- laritonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Several experiments have demonstrated measure- ment and control of the Berry curvature of exciton-polariton and photon bands in these systems [23–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Our work will focus on these Fabry-Perot cavity systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In this work, we theoretically propose a scheme for gener- ating topological polaritons that combines advantages of both arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='03287v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='chem-ph] 9 Jan 2023 2 the approaches mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Here, the light-matter cou- pling contains the non-trivial topology instead of the individ- ual photon or exciton components and optical pumping with circularly polarized light breaks TRS instead of a large mag- netic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Breaking TRS in a molecular system using the helicity of light is an idea that has been demonstrated in sev- eral other contexts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' it has been used to achieve all-optical non- reciprocity [27, 28] and theoretical results suggest that it can also induce optical-activity in achiral molecules [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Addi- tionally, a similar idea that relies on breaking TRS using cir- cularly polarized light has been previously proposed for po- lariton lattices by Bleu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We focus on the topological properties of polaritons formed by the coupling of Frenkel excitons hosted in organic semi- conductors with photon modes in a Fabry-Perot cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Here, optical pumping with circularly polarized light saturates cer- tain electronic transitions and breaks TRS in the system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' this results in non-zero Chern numbers of polariton bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We exploit the primary nonlinearity of organic exciton- polaritons, saturation [11], to generate topological exciton- polariton bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Our scheme relies on the contraction of Rabi splitting due to saturation, and we find modified Berry curva- ture and Chern number of the bands under circularly polarized pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The Berry curvature of the more photonic sections of the bands computed in our work can be experimentally measured using pump-probe spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Furthermore, the applicability of our scheme is not limited to organic polariton systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' It only requires certain key ingredients: transitions that can be selectively excited with circularly polarized light, saturation effects, and Rabi splitting contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' To highlight this, we compute the Berry curvature of two other systems un- der strong coupling and optical pumping: (a) Ce:YAG and (b) monolayer MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Our work provides a viable strategy to in- duce non-reciprocal behavior in standard microcavity polari- tons, leading to the optical tuning of isolators and circulators [27], as well as fabrication of elliptically-polarized lasers and condensates [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' RESULTS Model In our theoretical study, we consider a Fabry-Perot cavity containing a thin film of porphyrin molecules at the center and a bulk perylene crystal filling the rest of the volume (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The porphyrin and perylene molecules are not treated on an equal footing in our model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' while the molecular transitions of porphyrin are considered explicitly in the Hamiltonian, those of the perylene crystal are not, and they can be accounted for through effective cavity modes [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' This is a valid ap- proximation because we focus on photon modes with fre- quencies close to those of electronic transitions in porphyrin (∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='81eV) [32, 33] and far off-resonant from the transitions of perylene (∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='98eV) [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Here, the birefringent perylene crystal plays the role of providing anisotropy and emergent optical activity to the cavity modes [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We model each porphyrin molecule as a three-level elec- 1 |G⟩ |−!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' "#⟩ |+!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' "#⟩ 𝝁!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 𝝁" a b FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (a) Illustration of circularly polarized light exciting a met- alloporphyrin molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (b) Three-level model of porphyrin with a ground state |G⟩ and two degenerate excited states |+mol⟩,|−mol⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The transition dipole moment for a transition from |G⟩ to |±mol⟩ is µµµ± = µ0(ˆx±iˆy)/ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The number of yellow circles at each state rep- resents the fraction of molecules in that state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Here, the ratio of the fraction of molecules in the ground, fG, and |±mol⟩ excited states, f±, is fG : f+ : f− = 3 : 1 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Such population ratios can be achieved through pumping with circularly polarized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' tronic system with a ground state |G⟩ and two excited states |+mol⟩ and |−mol⟩ (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 2b) [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In the absence of a magnetic field, the two excited states are degenerate and the energy difference between the ground and excited states is ¯hωe = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='81eV [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The transition dipole moments for transi- tions from |G⟩ to |+mol⟩ and |−mol⟩ are µµµ+ = µ0(ˆx+iˆy)/ √ 2 and µµµ− = µ0(ˆx−iˆy)/ √ 2, respectively, with µ0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='84D [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Here, ˆx and ˆy are unit vectors along the x and y directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Using circular polarized light, the |+mol⟩ or |−mol⟩ states can be selectively excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In our model, we consider a thin film of metalloporphyrins or metallophtalocyanines arranged in a square lattice with nearest neighbor spacing a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The choice of lattice is irrele- vant because later we will take the continuum limit a → 0 as we are only interested in length scales much larger than the intermolecular spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Each molecule is labeled with the index m = (mx,my), where mx,my ∈ Z and the molecule’s position is given by rm = mxaˆx + myaˆy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' States of the mth molecule are then written as |m,G⟩, |m,+mol⟩ and |m,−mol⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The creation operator ˆσ† m,± = |m,±mol⟩⟨m,G| ⊗n̸=m In ex- cites the mth molecule from |m,G⟩ to |m,±mol⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Here, In = |n,G⟩⟨n,G| + |n,+mol⟩⟨n,+mol| + |n,−mol⟩⟨n,−mol| is the identity operator for nth molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' These molecular operators satisfy commutation relations (a generalization of the commu- tation relations of paulion operators [38, 39]), � ˆσn,±, ˆσ† m,± � = δm,n(1− ˆσ† n,∓ ˆσn,∓ −2 ˆσ† n,± ˆσn,±).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (1) We model the effective photon modes of a Fabry-Perot cav- ity filled with perylene as in Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [25] For the photon modes of a Fabry-Perot cavity, the component of wave vec- tor orthogonal to the mirrors kz = 2nπ/L is quantized, where L is the effective distance between the mirrors of the cav- ity and n is the mode index [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' For a given n, the modes are labeled by the in-plane wave vector k = kxˆx + kyˆy and polarization α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' the creation operators associated with these 00003 modes are ˆa† k,α and they satisfy bosonic commutation rela- tions � ˆak,α, ˆa† k′,α′ � = δα,α′δk,k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' As a result of in-plane trans- lational invariance of a cavity, k can take any value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=', kx,ky ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Throughout this work, we specify the cavity mode polarization in the circularly polarized basis α = ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The Hamiltonian of the full system is ˆH = ˆHmol + ˆHcav + ˆHcav−mol,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (2) where ˆHmol =∑ m � ¯hωe ˆσ† m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆσm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ + ¯hωe ˆσ† m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆσm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− � ˆHcav =∑ k �� E0 + ¯h2|k|2 2m∗ +ζ|k|cosφ � ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ + � E0 + ¯h2|k|2 2m∗ −ζ|k|cosφ � ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− + � −β0 +β|k|2e−i2φ� ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− + � −β0 +β|k|2ei2φ� ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHcav−mol =∑ m ∑ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α − ˆµµµm · ˆEk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α(rm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='0) ≈∑ m ∑ k eik·rm �NxNy � (µµµ+ ·Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+) ˆσ† m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ +(µµµ− ·Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+) ˆσ† m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ +(µµµ+ ·Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='−) ˆσ† m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− +(µµµ− ·Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='−) ˆσ† m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− � +H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (3) Above, ˆHmol describes the porphyrin molecules, ˆHcav the ef- fective cavity modes (including contributions from the pery- lene crystal), and ˆHcav−mol the coupling between the por- phyrin molecules and effective cavity modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Here, φ is the angle between the in-plane wave vector and the x-axis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=', cosφ = kx/|k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Within ˆHcav, β specifies the TE-TM splitting, β0 quantifies the linear birefringence of the perylene crystal which splits the H-V modes, and ζ describes the emergent optical activity [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Additionally, E0 is the frequency of the cavity modes at |k| = 0 in the absence of the perylene crys- tal (β0 = 0 and ζ = 0), and m∗ is the effective mass of the photons in the absence of perylene (β0 = 0 and ζ = 0) and TE-TM splitting (β = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The term ˆHmol describes an Nx ×××Ny array of porphyrin molecules with periodic boundary condi- tions along both the x and y directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We have made the electric dipole approximation and the rotating-wave approxi- mation in ˆHcav−mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Here, ˆµµµm is the electric dipole operator associated with the mth molecule and ˆEk,α(r,z) is the electric field operator of the mode with polarization α and in-plane wave vector k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In addition, µµµα′ · Jk,α is the collective cou- pling strength of the cavity mode labeled by k,α and the |G⟩ to ��α′ mol � transition of the molecules (see Supplementary sec- tion S1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The photon modes of an empty cavity experience TE-TM splitting due to polarization dependent reflection from the mir- rors [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' While the TE-TM splitting lifts the degeneracy be- tween photon modes at |k| ̸= 0, photon modes of both polar- izations remain degenerate at |k| = 0 due to rotational symme- try of the cavity mirrors about the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' However, for Berry curvature and Chern invariant to be well-defined, we need the photon/polariton bands to be separated in energy at all k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' to achieve this, we include the perylene crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The anisotropy and emergent optical activity of the perylene crystal lifts the degeneracy between the photon modes at all k [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' To compute the Berry curvature and Chern number, we fo- cus on the first excitation manifold which is spanned by states |m,±mol⟩ = ˆσ† m,± |vac⟩ and |k,±cav⟩ = ˆa† k,± |vac⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Here, |vac⟩ is the absolute ground state of the system where the pho- ton modes are empty and all molecules are in their ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Rewriting the Hamiltonian with operators ˆσk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' where ˆσm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α = 1 √ NxNy ∑k∈BZ eik·rm ˆσk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α and restricting ourselves to the first excitation manifold,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' we find ˆH(k) = ⟨k| ˆH |k⟩ to be ˆH(k) = ˆHmol(k)+ ˆHcav(k)+ ˆHcav−mol(k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (4) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHmol(k) =¯hωe |+mol⟩⟨+mol|+ ¯hωe |−mol⟩⟨−mol|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHcav(k) = � E0 + ¯h2|k|2 2m∗ +ζ|k|cosφ � |+cav⟩⟨+cav| + � E0 + ¯h2|k|2 2m∗ −ζ|k|cosφ � |−cav⟩⟨−cav| + � −β0 +β|k|2e−i2φ� |+cav⟩⟨−cav| + � −β0 +β|k|2ei2φ� |−cav⟩⟨+cav|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHcav−mol(k) =Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ · � µµµ+ |+mol⟩+ µµµ− |−mol⟩ � ⟨+cav| +Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− · � µµµ+ |+mol⟩+ µµµ− |−mol⟩ � ⟨−cav| +H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (5) Here, k lies within the first Brillouin zone determined by the porphyrin lattice kx,ky ∈ [−π/a,π/a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' As we are only inter- ested in length scales much larger than a, we take the contin- uum limit a → 0 while keeping µ0/a a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Therefore, terms such as the collective light-matter coupling strength, Jk,α · µµµα′, remain constant in this limit (see Supplementary section S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Moreover, upon taking the continuum limit, ˆH(k) does not change;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' only the range of k becomes infinitely large, kx,ky ∈ R, that is, our system acquires complete translational invariance in the x-y plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' For such continuous systems, since kx,ky ∈ R is unbounded, we need to map (kx,ky) onto a sphere which is a closed and bounded surface using stere- ographic projection before we compute Chern numbers [42] (see Supplementary section S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' When we diagonalize the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 5, we ob- tain four bands which we label with l = 1,2,3,4 in increas- ing order of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 3a we plot the Berry curvature, Ω1(k), of the lowest band l = 1, and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 3e we plot the ky = 0 slice of the band structure of the two bands lowest in energy, l = 1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' As expected, in the absence of optical pumping, this system preserves TRS, which can be verified 4 a e f g h b c d 𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 0 𝑓" = 0 𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3 𝑓" = 0 𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 0 𝑓" = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3 𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3 𝑓" = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3 S3 𝐶!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 0 𝐶" = 0 𝐶!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 1 𝐶" = −1 𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 0 𝑓" = 0 𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3 𝑓" = 0 𝐶!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = −1 𝐶" = 1 𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 0 𝑓" = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3 𝐶!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 0 𝐶" = 0 𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3 𝑓" = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3 Ω1 (𝜇m2) Ω1 (𝜇m2) Ω1 (𝜇m2) Ω1 (𝜇m2) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (a-d) Berry curvature of the lowest energy band, Ω1(k), and (e-h) a slice of the band structure at ky = 0 of the lower two bands, under different levels of optical pumping which create populations: (a,e) f+ = f− = 0, (b,f) f+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3, f− = 0, (c,g) f+ = 0, f− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3, and (d,h) f+ = f− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (e-h) The colors of the band indicate the value of the Stokes parameter, S3(k), which measures the degree of circular polarization of a mode (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The Chern numbers C1 and C2 of the bands are also specified and are non-zero under time-reversal symmetry (TRS) breaking, that is, when f+ ̸= f−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We used parameters β0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='1eV, β = 9×10−4eVµm2, ζ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='5×10−3eVµm, m∗ = 125¯h2eV−1µm−2, E0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='80eV and ¯hωe = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='81eV (see Supplementary section S4 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' using the condition on Berry curvature Ωl(k) = −Ωl(−k), and the Chern numbers of the all the bands Cl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Also, note that, the smallest splitting between the lower two bands within −13µm−1 < kx,ky < 13µm−1 is ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='8meV which is larger than the linewidth of the transition in porphyrin at 4K (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='5meV) [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Optical pumping Optical pumping can saturate the electronic transitions of a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' This leads to reduction in the effective light-matter coupling strength, and, therefore, Rabi splitting contraction [11, 45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' For instance, when the pump excites a fraction of molecules, fE, to the excited state and the remaining popu- lation stays in the ground state, fG, it results in Rabi splitting contraction proportional to √fG − fE = √1−2fE [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In our system, when the molecules are optically pumped, a fraction, f+, of the molecules occupy the |+mol⟩ state, an- other fraction, f−, occupy the |−mol⟩ state, and the remaining fraction, fG, are in the ground state |G⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The Rabi contraction corresponding to the |G⟩ to |+mol⟩ transition should then be proportional to √fG − f+ which equals √1− f− −2 f+ since fG+ f++ f− = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Similarly, the contraction should be propor- tional to √1− f+ −2 f− for the |G⟩ to |−mol⟩ transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' This difference in light-matter coupling when f+ ̸= f− effectively introduces 2D chirality into the system [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' To derive an effective Hamiltonian under optical pumping, we use Heisenberg equations of motion and make a mean- field approximation following the approach of Ribeiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [47] (Supplementary section S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We then obtain the effective Hamiltonian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHeff(k) = ˆHeff mol(k)+ ˆHeff cav(k)+ ˆHeff cav−mol(k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (6) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHeff mol(k) =¯hωe |+mol⟩′ ⟨+mol|′ + ¯hωe |−mol⟩′ ⟨−mol|′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHeff cav(k) = � E0 + ¯h2|k|2 2m∗ +ζ|k|cosφ � |+cav⟩′ ⟨+cav|′ + � E0 + ¯h2|k|2 2m∗ −ζ|k|cosφ � |−cav⟩′ ⟨−cav|′ + � −β0 +β|k|2e−i2φ� |+cav⟩′ ⟨−cav|′ + � −β0 +β|k|2ei2φ� |−cav⟩′ ⟨+cav|′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHeff cav−mol(k) =Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ · �� 1− f− −2 f+µµµ+ |+mol⟩′ + � 1− f+ −2 f−µµµ− |−mol⟩′ � ⟨+cav|′ +Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− · �� 1− f− −2 f+µµµ+ |+mol⟩′ + � 1− f+ −2 f−µµµ− |−mol⟩′ � ⟨−cav|′ +H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (7) Here, the states |γ⟩′ are different from states |γ⟩ in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 5, where γ = ±mol,±cav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' As expected, the light-matter coupling terms are scaled by factors √1− f∓ −2 f± which is a consequence of the commutation relation in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 1 (see Supplementary sec- tion S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' If the pump pulse is circularly polarized, f+ ̸= f−, the Rabi contraction factor that multiplies the light-matter coupling dif- fers for transitions to the |+mol⟩ and |−mol⟩ states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' as a re- sult, time-reversal symmetry is broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Consequently, when f+ > f−, we find that bands 1 and 2 have non-zero Chern numbers +1 and -1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 3f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Under the opposite condition, f+ < f−, the Chern numbers reverse sign as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 3g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' When f+ = f−, TRS is preserved, and all bands have Chern 5 a c d b 𝒇!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 𝟑 𝒇" = 𝟎 𝒇!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 𝟑 𝒇" = 𝟎 S3 Band 1 Band 2 𝒇!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 𝟎 𝒇" = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 𝟑 𝒇!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 𝟎 𝒇" = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 𝟑 Band 1 Band 2 S3 S3 S3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The Stokes parameter, S3(k), which is a measure of the degree of circular polarization of a mode (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 8), under pumping with (a,c) σ+ polarized light which creates populations f+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3, f− = 0 and (b,d) σ− polarized light which creates populations f+ = 0, f− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3 of the two lowest energy bands (Band 1 and 2 as indicated in the inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We used parameters β0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='1eV, β = 9×10−4eVµm2, ζ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='5 × 10−3eVµm, m∗ = 125¯h2eV−1µm−2, E0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='80eV and ¯hωe = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='81eV (see Supplementary section S4 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' number 0 as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 3e and 3h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 3b-c, we plot the computed Berry curvature when f+ ̸= f− and due to broken TRS, we find Ωl(k) ̸= −Ωl(−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We also plot the Stokes parameter, S3(k), for bands 1 and 2, under pumping with circularly polarized light, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The Stokes parameter, S3(k), provides information on the de- gree of circular polarization of the photonic component of an exciton-polariton band and is calculated as S3(k) = |b+,cav(k)|2 −|b−,cav(k)|2 |b+,cav(k)|2 +|b−,cav(k)|2 (8) where the eigenvectors of the band are ��ul,k � = b+,cav(k)|+cav⟩ + b−,cav(k)|−cav⟩ + b+,mol(k)|+mol⟩ + b−,mol(k)|−mol⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In the absence of pumping, we find that within a band, one half of the modes are predominantly σ+ polarized and the other half are σ− polarized (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 3e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Once TRS is broken with circularly polarized optical pumping, a large number of modes within each band become overwhelmingly of the same polarization (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 3f-g and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In experiments, the Berry curvature of photon bands in a Fabry-Perot cavity can be extracted from the components of the Stokes vector [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' However, in the case of exciton- polariton bands, the Berry curvature of only sections of the band that are predominantly photonic and have negligible molecular character [49] can be measured experimentally as, to the best of our knowledge, it is difficult to obtain the phase relationship between the photonic and molecular components, unless light-matter cross-correlation functions are measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Therefore, in our case, the Berry curvature of only parts of the band that are mostly photonic in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 3a-d can be measured using pump-probe spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' This measurement should be feasible as long as the time delay between the pump and probe pulses is shorter than the time the system takes to depo- larize and reach a state with f+ = f−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' As the depolarization timescale for porphyrins ranges from 210 fs to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='6 ps, this measurement should be viable [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' As the Chern numbers of bands 1 and 2 are modified through pumping with circularly polarized light, if we per- form a calculation where a region of the system is pumped with σ+ polarized light ( f+ ̸= 0 and f− = 0) and an adjacent region is pumped with σ− polarized light (f+ = 0 and f− ̸= 0), we expect edge states at the boundary between these regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' However, as our Hamiltonian does not contain couplings be- tween neighboring molecules, and the position of a molecule does not enter the Hamiltonian anywhere except through the phase of the light-matter coupling eik·rm, the standard bulk- boundary correspondence is no longer applicable and we do not observe edge states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We do not include plots for these cal- culations in this work and leave it an open question whether there is an analogous statement for bulk-boundary correspon- dence in these types of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' On the other hand, for exciton-polariton systems where nearest-neighbor couplings are present, edge states have been predicted and observed [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Other systems To emphasize that our scheme of saturating electronic tran- sitions with circularly polarized light to modify topological properties is not limited to organic exciton-polariton systems, we compute the Berry curvature of two other polariton sys- tems where porphyrin is replaced with (i) Ce:YAG and (ii) MoS2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 5a and 5d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Other materials can also be used in place of porphyrins, as long as they have transitions that can be selectively excited with circularly polarized light and these transitions have large enough transition dipole moments that they can couple strongly to the photon modes of a cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In Yttrium Aluminum garnet (YAG) doped with Cerium, Ce3+ ions replace some Y3+ and Ce3+ has transitions that can be selectively excited with circularly polarized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Here, each Ce3+ has two possible ground states, one with the elec- tron in spin up |4 f(1) ↑⟩, and the other with it in spin down |4 f(1) ↓⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Similarly, it has a degenerate pair of excited spin states |5d(1) ↑⟩ and |5d(1) ↓⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The |4 f(1) ↓⟩ ↔ |5d(1) ↑⟩ transition has ∼ 400 times larger oscillator strength for ex- citation with σ+ polarized light than with σ− polarized light, therefore, we take the transition dipole moment to be µµµ+ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 5b) [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Similarly, we take the transition dipole to be µµµ− for the |4f(1) ↑⟩ ↔ |5d(1) ↓⟩ transition (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The transitions in Ce:YAG do couple to photon modes, however, to the best of our knowledge, strong coupling has not been reported in the literature [52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nevertheless, strong light-matter cou- pling has been achieved with a similar system: Nd3+ doped YSO and YVO crystals [54, 55], and based on our calcula- tions, with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='1µm thick sample of Ce:YAG at concentration 1% Ce3+ (relative to Y3+), we should be able to attain strong 6 Ce:YAG a b d f e c |4f(1)↓⟩ |5d(1)↑⟩ |5d(1)↓⟩ 𝝁!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 𝝁" |4f(1)↑⟩ 𝑓↓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='4 𝑓↑ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='6 𝑓# = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3 𝑓#!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = 0 𝝁" 𝝁!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' K K’ Ω1 (𝜇m2) Ω1 (𝜇m2) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (a) Illustration of Ce:YAG (salmon block) and perylene (green blocks) within a Fabry-Perot cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (b) Atomic levels of Ce3+ ions embedded in Yttrium Aluminum garnet (YAG) where the yellow circles indicate the fraction f↓ of Ce3+ ions in the |4f(1) ↓⟩ state and the fraction f↑ in the |4f(1) ↑⟩ state after optical pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The transition dipoles µµµ± = µ0(ˆx ± iˆy)/ √ 2 are also indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (c) Berry curvature of the lowest energy band, Ω1(k), under pumping with circularly polarized which creates populations f↓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='4 and f↑ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (d) Illustration of monolayer MoS2 and perylene (green blocks) within a Fabry-Perot cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (e) Illustration of A-excitons in the K and K’ valleys of monolayer MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (f) Berry curvature of the lowest energy band, Ω1(k), under pumping with circularly polarized which creates exciton populations fK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3 and fK′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We used parameters β0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='1eV, β = 9×10−4eVµm2, ζ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='5×10−3eVµm, m∗ = 125¯h2eV−1µm−2, (c) E0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='50eV, ¯hωe = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='53eV and (f) E0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='80eV, ¯hωe = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='855eV (see Supplementary section S4 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' coupling with photon modes in a Fabry-Perot cavity (see Sup- plementary section S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Under thermal equilibrium, the populations of the |4f(1) ↑⟩ and |4 f(1) ↓⟩ states are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' However, under pumping with pulses of σ+ polarization, in the presence of a small magnetic field ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='049T, the population of |4 f(1) ↑⟩ will exceed that of |4 f(1) ↓⟩ because population is selectively removed from |4f(1) ↓⟩ and added to |5d(1) ↑⟩ by the circularly polarized pulses, but decay from the excited |5d(1) ↑⟩ state to the two ground states has equal probability [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In principle, a mag- netic field is not required;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' however, as we do not know the spin relaxation time in the absence of the magnetic field, we report the magnetic field used in the experimental study [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Under optical pumping with circularly polarized light, the 5d states will have very small populations which we take to be zero, while the |4f(1) ↓⟩ and |4 f(1) ↑⟩ states will have unequal populations f↓ and f↑, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' here, f↓ + f↑ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Op- tically pumped Ce:YAG can then be modeled using the effec- tive Hamiltonian in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 6 and 7, with |±mol⟩′ → |5d(1) ↑ / ↓⟩ and √1− f∓ −2 f± → � f↓/↑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The large spin relaxation time of ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='5 ms makes this system particularly well-suited for our scheme because it maintains f↓ ̸= f↑, and hence non-zero Chern invariants, for an extended period of time [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 5c we plot Berry curvature of the lowest band of a perylene filled cavity strongly coupled with Ce:YAG, where f↓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='4 and f↑ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='6 (see Supplementary section S4 for values of other parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' TMDs, such as single-layer MoS2, display optically con- trollable valley polarization and could also be used in place of porphyrins [57–59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Due to lack of inversion symme- try in these systems, the K and K’ valleys are inequivalent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' this results in optical selection rules that allow selective cre- ation of excitons at K and K’ valleys with σ+ and σ− polar- ized light, respectively [60, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Additionally, strong light- matter coupling has been observed when monolayer MoS2 is placed within a Fabry-Perot cavity [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' This system has depolarization times of ∼ 200fs - 5ps making it possible to measure Berry curvature using pump-probe spectroscopy be- fore depolarization occurs [62, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We model this exciton- polariton system (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 5d) using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 6 and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 7 (we focus on the A-exciton, see Supplementary section S4 for parame- ters) with |+mol⟩ → |K⟩, |−mol⟩ → |K′⟩ and √1− f∓ −2 f± → �1−2 fK/K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 5f we plot the Berry curvature of the lowest band when fK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='3 and fK′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Unfortunately, sig- nificant Rabi contraction upon optical pumping has not been experimentally observed in these systems which will make it challenging to observe Berry curvature as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 5f since our model relies on saturation effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' However, for exciton polaritons formed from monolayer TMDs, even if Rabi con- traction through resonant optical pumping may not produce the intended effect, off-resonant optical pumping can break the degeneracy of excitons in the K and K’ valleys through optical stark effect [64], and this may have interesting conse- quences for the Berry curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Additionally, if bilayer MoS2 is used in place of monolayer MoS2, effects on the Berry cur- vature described in our work may be more pronounced as bi- layer MoS2 hosts interlayer excitons which possess large op- tical nonlinearities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' specifically, they display saturation and 7 Rabi contraction under strong coupling [65, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Finally, so far we have only considered replacing porphyrin with a different material, such as MoS2 or Ce:YAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In addi- tion to this, perylene can also be replaced with other suitable materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In our work, we choose to use a cavity filled with perylene because we do not want degeneracy at any k within the photon bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Other systems also satisfy this requirement and could be used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' For instance, we could use an elec- trically tunable, highly anisotropic, liquid-crystal cavity with well separated H and V polarized photon modes [24, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' A perovskite cavity is another potential candidate due to its high anisotropy, and optical pumping may help lift the degeneracy of polariton modes in this system [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Additionally, other photonic structures can also be used instead of a cavity, as long as the photon bands are not degenerate at any k and have non-zero light-matter coupling at all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' CONCLUSION In summary, we show that TRS can be broken in organic exciton-polariton systems through selectively saturating elec- tronic transitions with a circularly polarized pump and that the resulting bands possess non-zero Chern invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In particu- lar, we demonstrate this theoretically for a Fabry-Perot cavity filled with porphyrin and perylene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The Berry curvature of the more photonic parts of the bands of this system can be measured experimentally using pump-probe spectroscopy, as long as the time delay is shorter than the depolarization time for porphyrin (210fs-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='6ps) [50], and this will reveal non-zero Berry curvature and Chern number under circularly polarized pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Our scheme relies on Rabi contraction from satu- ration of optical transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' It is important to note that edge states do not emerge in our system despite non-zero Chern in- variants as our model does not contain sufficient positional information about the molecules or the unit cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Bleu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [30] have previously proposed breaking TRS in inorganic exciton-polariton systems through pumping with circularly polarized light, however, their work relies on polariton con- densation and having patterned lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Finally, we demon- strate that saturating electronic transitions to modify topol- ogy is not limited to organic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' To illustrate this, we calculate the Berry curvature and Chern numbers of exciton- polariton bands of two other systems under optical pumping: (a) Ce:YAG and (b) monolayer MoS2, and find similar results as the organic exciton-polariton case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In view of recent devel- opments on electrically tuning the Berry curvature of liquid- crystal and perovskite filled cavities [24, 26], our work pro- vides an additional control knob to optically tune the Berry curvature of exciton-polariton systems using circularly polar- ized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Additionally, ultrafast control of topological prop- erties of systems with light may find use in nonreciprocal and nonlinear optoelectronic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ACKNOWLEDGEMENTS S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' acknowledges support from NSF Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' CA- REER CHE 1654732 for the development of the model and calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The conceptualization of the molecular and solid-state systems was guided by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' as part of the Center for Molecular Quantum Transduction (CMQT), an Energy Frontier Research Center funded by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Depart- ment of Energy, Office of Science, Basic Energy Sciences un- der Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' DE-SC0021314.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' thanks Kai Schwen- nicke and Stephan van den Wildenberg for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' CODE AVAILABILITY Code available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='com/SindhanaPS/Topological_Polaritons_Submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' REFERENCES [1] Claude Weisbuch, Mr Nishioka, A Ishikawa, and Y Arakawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Physical Review Letters, 69(23):3314, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [2] R André, D Heger, Le Si Dang, and Y Merle d’Aubigné.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Spec- troscopy of polaritons in cdte-based microcavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Journal of crystal growth, 184:758–762, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [3] David G Lidzey, DDC Bradley, MS Skolnick, T Virgili, S Walker, and DM Whittaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Strong exciton–photon coupling in an organic semiconductor microcavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nature, 395(6697): 53–55, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [4] R Butté, G Christmann, E Feltin, J-F Carlin, M Mosca, M Ilegems, and N Grandjean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Room-temperature polariton lu- minescence from a bulk gan microcavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Physical Review B, 73(3):033315, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [5] R Shimada, J Xie, Vitaliy Avrutin, Ü Özgür, and H Morkoˇc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Cavity polaritons in zno-based hybrid microcavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Applied Physics Letters, 92(1):011127, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [6] Antoine Brehier, Radoslav Parashkov, Jean-Sébastien Lauret, and Emmanuelle Deleporte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Strong exciton-photon coupling in a microcavity containing layered perovskite semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Applied physics letters, 89(17):171110, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [7] Rui Su, Antonio Fieramosca, Qing Zhang, Hai Son Nguyen, Emmanuelle Deleporte, Zhanghai Chen, Daniele Sanvitto, Tim- othy CH Liew, and Qihua Xiong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Perovskite semiconductors for room-temperature exciton-polaritonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nature Materials, 20(10):1315–1324, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [8] Xiaoze Liu, Tal Galfsky, Zheng Sun, Fengnian Xia, Erh- chen Lin, Yi-Hsien Lee, Stéphane Kéna-Cohen, and Vinod M Menon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Strong light–matter coupling in two-dimensional atomic crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nature Photonics, 9(1):30–34, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [9] Fengrui Hu and Zhe Fei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Recent progress on exciton polaritons in layered transition-metal dichalcogenides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Advanced Optical Materials, 8(5):1901003, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [10] James A Hutchison, Tal Schwartz, Cyriaque Genet, Eloïse De- vaux, and Thomas W Ebbesen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Modifying chemical landscapes by coupling to vacuum fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Angewandte Chemie Interna- tional Edition, 51(7):1592–1596, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [11] KS Daskalakis, SA Maier, Ray Murray, and Stéphane Kéna- Cohen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nonlinear interactions in an organic polariton conden- sate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nature materials, 13(3):271–278, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 8 [12] Christof P Dietrich, Anja Steude, Laura Tropf, Marcel Schu- bert, Nils M Kronenberg, Kai Ostermann, Sven Höfling, and Malte C Gather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' An exciton-polariton laser based on biolog- ically produced fluorescent protein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Science advances, 2(8): e1600666, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [13] Nina Krainova, Alex J Grede, Demetra Tsokkou, Natalie Banerji, and Noel C Giebink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Polaron photoconductivity in the weak and strong light-matter coupling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Physical review letters, 124(17):177401, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [14] Qing Liao, Charly Leblanc, Jiahuan Ren, Feng Li, Yiming Li, Dmitry Solnyshkov, Guillaume Malpuech, Jiannian Yao, and Hongbing Fu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Experimental measurement of the divergent quantum metric of an exceptional point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Physical Review Let- ters, 127(10):107402, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [15] Marco Dusel, Simon Betzold, Tristan H Harder, Monika Em- merling, Johannes Beierlein, Jürgen Ohmer, Utz Fischer, Ronny Thomale, Christian Schneider, Sven Hofling, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Room- temperature topological polariton laser in an organic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nano Letters, 21(15):6398–6405, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [16] Dmitry D Solnyshkov, Guillaume Malpuech, Philippe St-Jean, Sylvain Ravets, Jacqueline Bloch, and Alberto Amo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Micro- cavity polaritons for topological photonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Optical Materials Express, 11(4):1119–1142, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [17] Charles-Edouard Bardyn, Torsten Karzig, Gil Refael, and Tim- othy CH Liew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Topological polaritons and excitons in garden- variety systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Physical Review B, 91(16):161413, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [18] Joel Yuen-Zhou, Semion K Saikin, Tony Zhu, Mehmet C On- basli, Caroline A Ross, Vladimir Bulovic, and Marc A Baldo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Plexciton dirac points and topological modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nature commu- nications, 7(1):1–7, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [19] S Klembt, TH Harder, OA Egorov, K Winkler, R Ge, MA Ban- dres, M Emmerling, L Worschech, TCH Liew, M Segev, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Exciton-polariton topological insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nature, 562(7728): 552–556, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [20] Torsten Karzig, Charles-Edouard Bardyn, Netanel H Lindner, and Gil Refael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Topological polaritons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Physical Review X, 5 (3):031001, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [21] Wenjing Liu, Zhurun Ji, Yuhui Wang, Gaurav Modi, Minsoo Hwang, Biyuan Zheng, Volker J Sorger, Anlian Pan, and Ritesh Agarwal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Generation of helical topological exciton-polaritons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Science, 370(6516):600–604, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [22] Mengyao Li, Ivan Sinev, Fedor Benimetskiy, Tatyana Ivanova, Ekaterina Khestanova, Svetlana Kiriushechkina, Anton Vaku- lenko, Sriram Guddala, Maurice Skolnick, Vinod M Menon, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Experimental observation of topological z2 exciton- polaritons in transition metal dichalcogenide monolayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Na- ture communications, 12(1):1–10, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [23] A Gianfrate, O Bleu, L Dominici, V Ardizzone, M De Giorgi, D Ballarini, G Lerario, KW West, LN Pfeiffer, DD Solnyshkov, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Measurement of the quantum geometric tensor and of the anomalous hall drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nature, 578(7795):381–385, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [24] Katarzyna Rechci´nska, Mateusz Król, Rafał Mazur, Prze- mysław Morawiak, Rafał Mirek, Karolina Łempicka, Witold Bardyszewski, Michał Matuszewski, Przemysław Kula, Wik- tor Piecek, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Engineering spin-orbit synthetic hamiltonians in liquid-crystal optical cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Science, 366(6466):727–730, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [25] Jiahuan Ren, Qing Liao, Feng Li, Yiming Li, Olivier Bleu, Guillaume Malpuech, Jiannian Yao, Hongbing Fu, and Dmitry Solnyshkov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nontrivial band geometry in an optically active system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nature communications, 12(1):1–8, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [26] Karolina Łempicka-Mirek, Mateusz Król, Helgi Sigurdsson, Adam Wincukiewicz, Przemysław Morawiak, Rafał Mazur, Marcin Muszy´nski, Wiktor Piecek, Przemysław Kula, Tomasz Stefaniuk, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Electrically tunable berry curvature and strong light-matter coupling in liquid crystal microcavities with 2d perovskite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Science Advances, 8(40):eabq7533, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [27] Sriram Guddala, Yuma Kawaguchi, Filipp Komissarenko, Svet- lana Kiriushechkina, Anton Vakulenko, Kai Chen, Andrea Alù, Vinod M Menon, and Alexander B Khanikaev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' All-optical nonreciprocity due to valley polarization pumping in transi- tion metal dichalcogenides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nature communications, 12(1):1–9, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [28] Erik J Lenferink, Guohua Wei, and Nathaniel P Stern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Coher- ent optical non-reciprocity in axisymmetric resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Optics express, 22(13):16099–16111, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [29] Kai Schwennicke and Joel Yuen-Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Optical activity from the exciton aharonov–bohm effect: A floquet engineering ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The Journal of Physical Chemistry C, 124(7):4206– 4214, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [30] O Bleu, DD Solnyshkov, and Guillaume Malpuech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Photonic versus electronic quantum anomalous hall effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Physical Re- view B, 95(11):115415, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [31] Teng Long, Xuekai Ma, Jiahuan Ren, Feng Li, Qing Liao, Stefan Schumacher, Guillaume Malpuech, Dmitry Solnyshkov, and Hongbing Fu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Helical polariton lasing from topological val- leys in an organic crystalline microcavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Advanced Science, 9 (29):2203588, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [32] Mercedes Rubio, Björn O Roos, Luis Serrano-Andrés, and Manuela Merchán.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Theoretical study of the electronic spectrum of magnesium-porphyrin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The Journal of chemical physics, 110 (15):7202–7209, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [33] Lawrence Edwards, David H Dolphin, and Martin Gouterman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Porphyrins: Xvi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' vapor absorption spectra and redox reactions: Octalkylporphins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Journal of Molecular Spectroscopy, 35(1): 90–109, 1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [34] Tonatiuh Rangel, Andre Rinn, Sahar Sharifzadeh, Felipe H da Jornada, André Pick, Steven G Louie, Gregor Witte, Leeor Kronik, Jeffrey B Neaton, and Sangam Chatterjee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Low-lying excited states in crystalline perylene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Proceedings of the Na- tional Academy of Sciences, 115(2):284–289, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [35] Ingo Barth, Jörn Manz, Yasuteru Shigeta, and Kiyoshi Yagi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Unidirectional electronic ring current driven by a few cycle cir- cularly polarized laser pulse: quantum model simulations for mg- porphyrin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Journal of the American Chemical Society, 128 (21):7043–7049, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [36] Joel Yuen-Zhou, Semion K Saikin, Norman Y Yao, and Alán Aspuru-Guzik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Topologically protected excitons in porphyrin thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nature materials, 13(11):1026–1032, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [37] Shichao Sun, Bing Gu, and Shaul Mukamel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Polariton ring cur- rents and circular dichroism of mg-porphyrin in a chiral cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Chemical science, 13(4):1037–1048, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [38] Shaul Mukamel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Principles of nonlinear optical spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Number 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Oxford University Press on Demand, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [39] Vladimir M Agranovich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Excitations in organic solids, volume 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' OUP Oxford, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [40] Alexey V Kavokin, Jeremy J Baumberg, Guillaume Malpuech, and Fabrice P Laussy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Microcavities, volume 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Oxford uni- versity press, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [41] Giovanna Panzarini, Lucio Claudio Andreani, A Armitage, D Baxter, MS Skolnick, VN Astratov, JS Roberts, Alexey V Kavokin, Maria R Vladimirova, and MA Kaliteevski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Exciton- light coupling in single and coupled semiconductor microcav- ities: Polariton dispersion and polarization splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Physical Review B, 59(7):5082, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [42] Mário G Silveirinha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Chern invariants for continuous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Physical Review B, 92(12):125153, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 9 [43] Uzi Even, Jacob Magen, Joshua Jortner, Joel Friedman, and Haim Levanon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Isolated ultracold porphyrins in super- sonic expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' free-base tetraphenylporphyrin and zn- tetraphenylporphyrin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The Journal of Chemical Physics, 77(9): 4374–4383, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [44] S Voelker, RM Macfarlane, AZ Genack, HP Trommsdorff, and JH van Der Waals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Homogeneous linewidth of the s 1← s 0 transition of free-base porphyrin in an n-octane crystal as stud- ied by photochemical hole-burning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The Journal of Chemical Physics, 67(4):1759–1765, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [45] Bo Xiang, Raphael F Ribeiro, Adam D Dunkelberger, Jiaxi Wang, Yingmin Li, Blake S Simpkins, Jeffrey C Owrutsky, Joel Yuen-Zhou, and Wei Xiong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Two-dimensional infrared spec- troscopy of vibrational polaritons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences, 115(19):4845–4850, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [46] Timur Yagafarov, Denis Sannikov, Anton Zasedatelev, Kyriacos Georgiou, Anton Baranikov, Oleksandr Kyriienko, Ivan She- lykh, Lizhi Gai, Zhen Shen, David Lidzey, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Mechanisms of blueshifts in organic polariton condensates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Communications Physics, 3(1):1–10, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [47] Raphael F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Ribeiro, Adam D Dunkelberger, Bo Xiang, Wei Xiong, Blake S Simpkins, Jeffrey C Owrutsky, and Joel Yuen- Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Theory for nonlinear spectroscopy of vibrational polari- tons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The journal of physical chemistry letters, 9(13):3766– 3771, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [48] Andrew H Salij, Randall H Goldsmith, and Roel Tempelaar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Chiral polaritons based on achiral fabry-perot cavities using ap- parent circular dichroism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' arXiv preprint arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='14461, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [49] Laura Polimeno, Giovanni Lerario, Milena De Giorgi, Luisa De Marco, Lorenzo Dominici, Francesco Todisco, Annalisa Coriolano, Vincenzo Ardizzone, Marco Pugliese, Carmela T Prontera, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Tuning of the berry curvature in 2d perovskite polaritons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nature nanotechnology, 16(12):1349–1354, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [50] C Galli, Klaas Wynne, Steven M LeCours, MJ Therien, and RM Hochstrasser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Direct measurement of electronic dephasing using anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Chemical physics letters, 206(5-6):493–499, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [51] Roman Kolesov, Kangwei Xia, Rolf Reuter, Mohammad Ja- mali, Rainer Stöhr, Tugrul Inal, Petr Siyushev, and Jörg Wrachtrup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Mapping spin coherence of a single rare-earth ion in a crystal onto a single photon polarization state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Physical review letters, 111(12):120502, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [52] Robert J Moerland, I Gerward C Weppelman, Marijke Scotuzzi, and Jacob P Hoogenboom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nanoscale imaging of light-matter coupling inside metal-coated cavities with a pulsed electron beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nano Letters, 18(10):6107–6112, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [53] SRK Rodriguez, S Murai, MA Verschuuren, and J Gómez Ri- vas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Light-emitting waveguide-plasmon polaritons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Physical review letters, 109(16):166803, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [54] Tian Zhong, Jonathan M Kindem, Evan Miyazono, and Andrei Faraon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nanophotonic coherent light–matter interfaces based on rare-earth-doped crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nature communications, 6(1):1– 6, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [55] Tian Zhong, Jonathan M Kindem, Jake Rochman, and An- drei Faraon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Interfacing broadband photonic qubits to on-chip cavity-protected rare-earth ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nature communications, 8(1):1–7, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [56] P Siyushev, K Xia, R Reuter, M Jamali, N Zhao, N Yang, C Duan, N Kukharchyk, AD Wieck, R Kolesov, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Coherent properties of single rare-earth spin qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nature communica- tions, 5(1):1–6, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [57] Kin Fai Mak, Keliang He, Jie Shan, and Tony F Heinz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Control of valley polarization in monolayer mos2 by optical helicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nature nanotechnology, 7(8):494–498, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [58] Hualing Zeng, Junfeng Dai, Wang Yao, Di Xiao, and Xiaodong Cui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Valley polarization in mos2 monolayers by optical pump- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nature nanotechnology, 7(8):490–493, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [59] Aswini Kumar Pattanayak, Pritam Das, Devarshi Chakrabarty, Avijit Dhara, Shreya Paul, Satyait Maji, Maruthi Manoj Brun- davanam, and Sajal Dhara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Probing spin dynamics of 2d ex- citons with twisted light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ACS Photonics, 9(10):3351–3356, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [60] Liuyang Sun, Chun-Yuan Wang, Alex Krasnok, Junho Choi, Jinwei Shi, Juan Sebastian Gomez-Diaz, André Zepeda, Shangjr Gwo, Chih-Kang Shih, Andrea Alù, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Separation of valley excitons in a mos2 monolayer using a subwavelength asymmetric groove array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nature Photonics, 13(3):180–184, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [61] Guan-Hao Peng, Oscar Javier Gomez Sanchez, Wei-Hua Li, Ping-Yuan Lo, and Shun-Jen Cheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Twisted-light-induced ex- citon wave packets in transition-metal dichalcogenide monolay- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='02081, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [62] Stefano Dal Conte, Federico Bottegoni, Eva Arianna Aurelia Pogna, D De Fazio, Stefano Ambrogio, Ilaria Bargigia, Cosimo D’Andrea, A Lombardo, M Bruna, Franco Ciccacci, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Ul- trafast valley relaxation dynamics in monolayer mos 2 probed by nonequilibrium optical techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Physical Review B, 92 (23):235425, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [63] Yen-Jung Chen, Jeffrey D Cain, Teodor K Stanev, Vinayak P Dravid, and Nathaniel P Stern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Valley-polarized exciton– polaritons in a monolayer semiconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nature Photonics, 11(7):431–435, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [64] Trevor LaMountain, Jovan Nelson, Erik J Lenferink, Samuel H Amsterdam, Akshay A Murthy, Hongfei Zeng, Tobin J Marks, Vinayak P Dravid, Mark C Hersam, and Nathaniel P Stern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Valley-selective optical stark effect of exciton-polaritons in a monolayer semiconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nature communications, 12(1):1–7, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [65] Biswajit Datta, Mandeep Khatoniar, Prathmesh Desh- mukh, Félix Thouin, Rezlind Bushati, Simone De Liberato, Stephane Kena Cohen, and Vinod M Menon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Highly non- linear dipolar exciton-polaritons in bilayer mos2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nature communications, 13(1):1–7, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [66] Charalambos Louca, Armando Genco, Salvatore Chiavazzo, Thomas P Lyons, Sam Randerson, Chiara Trovatello, Pe- ter Claronino, Rahul Jayaprakash, Kenji Watanabe, Takashi Taniguchi, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nonlinear interactions of dipolar excitons and polaritons in mos2 bilayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' arXiv preprint arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='00485, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [67] Marcin Muszy´nski, Mateusz Król, Katarzyna Rechci´nska, Przemysław Oliwa, Mateusz K˛edziora, Karolina Łempicka- Mirek, Rafał Mazur, Przemysław Morawiak, Wiktor Piecek, Przemysław Kula, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Realizing persistent-spin-helix las- ing in the regime of rashba-dresselhaus spin-orbit coupling in a dye-filled liquid-crystal optical microcavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Physical Review Applied, 17(1):014041, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Molecular and solid-state topological polaritons via optical saturation: supplemental document Sindhana Pannir-Sivajothi,1 Nathaniel P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Stern,2 and Joel Yuen-Zhou1, ∗ 1Department of Chemistry and Biochemistry, University of California San Diego, La Jolla, California 92093, USA 2Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' LIGHT-MATTER COUPLING The light-matter coupling part of the total Hamiltonian under the electric dipole approximation is, ˆHcav−mol =∑ m ∑ k,α − ˆµµµm · ˆEk,α(rm,0), =∑ m ∑ k,α − � ∑ α′=± (µµµα′ ˆσ† m,α′ + µµµ∗ α′ ˆσm,α′) � ˆEk,α(rm,0), (S1) where µµµα′ = µµµm,α′ = � m,α′ mol �� ˆµµµ |m,G⟩ is independent of m since we assume that all porphyrin molecules lie flat in the x-y plane and are oriented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The electric field operator of the mode labeled by k and α is ˆEk,α(r,z) = � ¯hωk,α 2Vεε0 � f∗ k,α(r,z) ˆa† k,α +fk,α(r,z) ˆak,α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (S2) Here, V = LxLyLz is the volume of the box we consider, where as mentioned in the main manuscript, we apply periodic boundary conditions along the x and y directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' From here on, we will call the in-plane area of the box A = LxLy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Here, fk,α(r,z) is the mode profile and it satisfies[1] � dr � Lz 0 dzf∗ k,α(r,z)fk,α(r,z) = LzA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (S3) For the TE and TM modes[2], fk,TE(r,z) =eik·r√ 2sin � nzπ Lz � z+ Lz 2 �� ˆφφφ, fk,TM(r,z) =eik·r � 2 |k|2 + � nzπ Lz �2 ��nzπ Lz � sin � nzπ Lz � z+ Lz 2 �� ˆρρρ −i|k|cos � nzπ Lz � z+ Lz 2 �� ˆz � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (S4) We make the rotating-wave approximation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHcav−mol =∑ m ∑ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α − � ∑ α′=± (µµµα′ ˆσ† m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α′ + µµµ∗ α′ ˆσm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α′) � �� ¯hωk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α 2Vεε0 � f∗ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α(rm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='0) ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α +fk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α(rm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='0) ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ≈ ∑ m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α′ ∑ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α − � ¯hωk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α 2Vεε0 � µµµα′ ·fk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α(rm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='0) ˆσ† m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α′ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α + µµµ∗ α′ ·f∗ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α(rm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='0) ˆσm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α′ ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' = ∑ m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α′ ∑ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α � eik·rm �NxNy (µµµα′ ·Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α) ˆσ† m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α′ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α + e−ik·rm �NxNy (µµµ∗ α′ ·J∗ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α) ˆσm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α′ ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (S5) where Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α = −�NxNy � ¯hωk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α 2Vεε0 e−ik·rfk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='0) and µµµα′ ·Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='α is the collective light-matter coupling strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The annihilation operators of photon modes polarized along the horizontal (H) or x-axis and vertical (V) or y-axis are ˆak,H and ˆak,V, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' They are related to α = ± polarized modes through ˆak,± = 1 √ 2( ˆak,H ∓i ˆak,V)[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' In addition, we assume ∗ joelyuen@ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='edu arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='03287v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='chem-ph] 9 Jan 2023 2 that they are related to the TM and TE modes through ˆak,TM = cosφ ˆak,H +sinφ ˆak,V and ˆak,TE = −sinφ ˆak,H +cosφ ˆak,V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Using this, we obtain the relationship between ˆak,TE, ˆak,TM and ˆak,+, ˆak,− modes to be, ˆak,TM = 1 √ 2 � eiφ ˆak,+ +e−iφ ˆak,− � , ˆak,TE = 1 √ 2 � ieiφ ˆak,+ −ie−iφ ˆak,− � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (S6) It is important to note that, based on these relationships and S4, the α =H/V modes are not completely linearly polarized and the α = ± modes are not completely circularly polarized when |k| becomes comparable with nzπ/Lz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We also find, Jk,+ = eiφ √ 2 � Jk,TM +iJk,TE � , Jk,− =e−iφ √ 2 � Jk,TM −iJk,TE � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (S7) To keep the collective coupling strength µµµα′ ·Jk,α constant while taking the a → 0 limit, we take the magnitude of the collective transition dipole of the bright state �NxNyµ0 over square root of the quantization area of the photon mode √ A to be a constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' that is, we keep √ρAµ0 = µ0/a a constant, where ρA = NxNy/A is the areal density of quantum emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Jk,α =−√ρA � ¯hωk,α 2Lzεε0 e−ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='rfk,α(r,0) =− 1 a � ¯hωk,α 2Lzεε0 e−ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='rfk,α(r,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (S8) S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' CHERN NUMBER CALCULATION a b (kmax,kmax) (kmax,-kmax) (-kmax,-kmax) (-kmax,kmax) x y FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (a) This is a cartoon figure that demonstrates the way Berry flux and Chern number are computed in our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The small squares are the plaquettes over which Berry flux is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The blue arrows specify the orientation used for Berry flux computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Note that the direction is opposite for the small squares and the large square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (b) Same as (a), but placed on a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Here, it is more clear that the direction of the arrow for the large square indicates the way Berry flux is computed for the giant plaquette covering the rest of the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' For the Chern invariant to be an integer, it is important that the Berry curvature is integrated over a closed and bounded surface [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' For periodic systems with a finite period, the Brillouin zone is a torus which satisfies this requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' However, 3 for a continuous system, (kx,ky) lies on an unbounded plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' for such systems, Silveirinha[5] proposed mapping this infinitely large plane onto a sphere to compute the Chern number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' This is the procedure we follow in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We discretize k-space and compute the Berry flux in each plaquette within a square-shaped region in k-space, −kmax ≤ kx,ky ≤ kmax [4, 6] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' S1a and S1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The entire region that satisfies the condition kx,ky > kmax or kx,ky < −kmax is taken as a single giant plaquette (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' S1b), and the Berry flux within this region is computed by taking the Berry phase along the boundary of the plaquette but in a direction opposite to that used to compute Berry flux for plaquettes within the square −kmax ≤ kx,ky ≤ kmax as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' S1a and S1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' To ensure that we obtain a converged Chern number, we calculate the Chern number for different kmax and find that, for our system, once kmax ≳ 100µm−1, the Chern number converges to C1 = ±1,C2 = ∓1,C3 = 0, and C4 = 0 when f+ ̸= f− with |f+ − f−| ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Smaller differences between f+ and f−, |f+ − f−| ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='11 require larger kmax for convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' This is not a problem for the f+ = f− case because the Chern invariant will always be zero due to time-reversal symmetry Ωl(k) = −Ωl(−k), and we can use kmax ≈ 100µm−1 to compute it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' OPTICAL PUMPING The number of excitations in the system Nex = ∑k,α a† k,αak,α + ∑n,α σ† n,ασn,α is a conserved quantity of this Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Therefore, when we have f+ fraction of molecules in the |+mol⟩ state and f− in the |−mol⟩ state, we will only have to look at the ( f+ + f−)Nth excitation manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Unfortunately, the dimensions of the Hilbert space of this manifold scale as � N (f++f−)N � , and this quickly becomes computationally intractable as the system size, N, increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Using mean-field theory, we reduce this many-body problem to a one-body problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' That is, we derive an effective Hamiltonian for a single excitation in the mean-field of the remaining ( f+ + f−)N excitations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' in this way, we reduce the dimensions of the Hilbert space to that of the first excitation manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' To do this, we follow a procedure similar to that used by Ribeiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [7] and write the Heisenberg equations of motion (EOM) for the operators ˆσm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='± and ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' i¯hd ˆσn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='± dt = � ˆσn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHmol � + � ˆσn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHcav � + � ˆσn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHcav−mol � =¯hωe ˆσn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='± + 1 �NxNy ∑ k eik·rn � (1− ˆσ† n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='∓ ˆσn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='∓ −2 ˆσ† n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='± ˆσn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='±) � Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ · µµµ± ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ +Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− · µµµ± ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− � − ˆσ† n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='∓ ˆσn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='± � Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ · µµµ∓ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ +Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− · µµµ∓ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' i¯hd ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='± dt = � ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHmol � + � ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHcav � + � ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHcav−mol � = � E0 + ¯h2|k|2 2m∗ ±ζ|k|cosφ � ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='± + � −β0 +β|k|2e∓i2φ� ˆa∓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='k + 1 �NxNy ∑ m eik·rm � J∗ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='± · µµµ∗ + ˆσm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ +J∗ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='± · µµµ∗ − ˆσm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (S9) We make a mean-field approximation to linearize these EOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' For instance, we use mn ≈ ¯mn, that is, ˆσ† n,+ ˆσn,+ ˆak,+ = � ⟨ ˆσ† n,+ ˆσn,+⟩+ ˆσ† n,+ ˆσn,+ −⟨ ˆσ† n,+ ˆσn,+⟩ � ˆak,+ =⟨ ˆσ† n,+ ˆσn,+⟩ ˆak,+ +( ˆσ† n,+ ˆσn,+ −⟨ ˆσ† n,+ ˆσn,+⟩)⟨ ˆak,+⟩ ≈⟨ ˆσ† n,+ ˆσn,+⟩ ˆak,+, (S10) where ⟨ ˆO⟩ = Tr � ˆρ0 ˆO � with ˆρ0 ≈ ∏m ˆρm ∏k ∏α=+,− ˆρα,k[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Here, we assume that after dephasing of the molecular amplitudes, ˆρm = fG |m,G⟩⟨m,G|+ f+ |m,+mol⟩⟨m,+mol|+ f− |m,−mol⟩⟨m,−mol|, ˆρα,k = |k,αcav,0⟩⟨k,αcav,0|, and, therefore, ⟨ ˆak,+⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The EOM then become i¯hd ˆσn,± dt ≈¯hωe ˆσn,± + 1 �NxNy (1− f∓ −2 f±)∑ k eik·rn � Jk,+ · µµµ± ˆak,+ +Jk,− · µµµ± ˆak,− � , i¯hd ˆak,± dt = � E0 + ¯h2|k|2 2m∗ ±ζ|k|cosφ � ˆak,± + � −β0 +β|k|2e∓i2φ� ˆa∓,k + 1 �NxNy ∑ m eik·rm � J∗ k,± · µµµ∗ + ˆσm,+ +J∗ k,± · µµµ∗ − ˆσm,− � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (S11) 4 We define rescaled operators ˆσ′ n,± = ˆσn,±/√1− f∓ −2 f± and rewrite the EOM, i¯hd ˆσ′ n,± dt ≈¯hωe ˆσ′ n,± + 1 �NxNy � 1− f∓ −2f±∑ k eik·rn � Jk,+ · µµµ± ˆak,+ +Jk,− · µµµ± ˆak,− � , i¯hd ˆak,± dt = � E0 + ¯h2|k|2 2m∗ ±ζ|k|cosφ � ˆak,± + � −β0 +β|k|2e∓i2φ� ˆa∓,k + �NxNy ∑ m eik·rm �� 1− f− −2f+J∗ k,± · µµµ∗ + ˆσ′ m,+ + � 1− f+ −2 f−J∗ k,± · µµµ∗ − ˆσ′ m,− � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (S12) From these EOM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' along with the fact that ˆσ′ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='± act effectively as bosonic operators in mean-field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' � ˆσ′ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆσ′† n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ � = 1− ˆσ† n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆσn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='−−2 ˆσ† n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆσn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ 1−f−−2 f+ ≈ ˆI and � ˆσ′ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆσ′† n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− � = − ˆσ† n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆσn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ 1−f−−2 f+ ≈ ˆ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' where ˆI and ˆ0 are the identity and zero operators,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' we can construct an effective Hamiltonian ˆHeff = ˆHeff mol + ˆHeff cav + ˆHeff cav−mol in ˆσ′ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='± and ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHeff mol =∑ n � ¯hωe ˆσ′† n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆσ′ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ + ¯hωe ˆσ′† n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆσ′ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHeff cav =∑ k � E0 + ¯h2|k|2 2m∗ +ζ|k|cosφ � ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ + � E0 + ¯h2|k|2 2m∗ −ζ|k|cosφ � ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− + � −β0 +β|k|2e−i2φ� ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− + � −β0 +β|k|2ei2φ� ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHeff cav−mol = 1 �NxNy ∑ m ∑ k eik·rm � � 1− f− −2 f+ � Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ · µµµ+ ˆσ′† m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ +Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− · µµµ+ ˆσ′† m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− � + � 1− f+ −2f− � Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ · µµµ− ˆσ′† m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ +Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− · µµµ− ˆσ′† m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− �� +H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=', (S13) which is the mean-field Hamiltonian when the system has f+, f− excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Writing this effective Hamiltonian in k-space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHeff mol =∑ k � ¯hωe ˆσ′† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆσ′ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ + ¯hωe ˆσ′† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆσ′ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHeff cav =∑ k � E0 + ¯h2|k|2 2m∗ +ζ|k|cosφ � ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ + � E0 + ¯h2|k|2 2m∗ −ζ|k|cosφ � ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− + � −β0 +β|k|2e−i2φ� ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− + � −β0 +β|k|2ei2φ� ˆa† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHeff cav−mol =∑ k � � 1− f− −2f+ � Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ · µµµ+ ˆσ′† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ +Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− · µµµ+ ˆσ′† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− � + � 1− f+ −2f− � Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ · µµµ− ˆσ′† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ +Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− · µµµ− ˆσ′† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− ˆak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− �� +H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (S14) We define states |k,±mol⟩′ and |k,±cav⟩′ corresponding to operators ˆσ′† k,± and ˆa† k,±, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Writing the Hamiltonian ˆHeff(k) = ⟨k| ˆHeff |k⟩ in the above basis we obtain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHeff(k) = ˆHeff mol(k)+ ˆHeff cav(k)+ ˆHeff cav−mol(k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (S15) 5 where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHeff mol(k) =¯hωe |+mol⟩′ ⟨+mol|′ + ¯hωe |−mol⟩′ ⟨−mol|′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHeff cav(k) = � E0 + ¯h2|k|2 2m∗ +ζ|k|cosφ � |+cav⟩′ ⟨+cav|′ + � E0 + ¯h2|k|2 2m∗ −ζ|k|cosφ � |−cav⟩′ ⟨−cav|′ + � −β0 +β|k|2e−i2φ� |+cav⟩′ ⟨−cav|′ + � −β0 +β|k|2ei2φ� |−cav⟩′ ⟨+cav|′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' ˆHeff cav−mol(k) =Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='+ · �� 1− f− −2 f+µµµ+ |+mol⟩′ + � 1− f+ −2 f−µµµ− |−mol⟩′ � ⟨+cav|′ +Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='− · �� 1− f− −2 f+µµµ+ |+mol⟩′ + � 1− f+ −2f−µµµ− |−mol⟩′ � ⟨−cav|′ +H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' (S16) S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' PARAMETERS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Perylene filled cavity We take parameters for the perylene filled cavity β0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='1eV, β = 9×10−4eVµm2, ζ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='5×10−3eVµm, m∗ = 125¯h2eV−1µm−2, and Lz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='745µm, where these are similar to those used to model the experiments of Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [9] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 3, 4, and 5 in main manuscript).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' On the other hand, we modify E0 and nz such that they make the photon modes in our model near resonant with the transition that is strongly coupled to the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' For instance, we take E0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='80eV and nz = 11 for porphyrin (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 3 and 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' E0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='50eV and nz = 9 for Ce:YAG (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 5b-c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' and E0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='80eV and nz = 5 for MoS2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 5e-f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We assume that perylene has a similar effect on these different photon modes, as it does on modes with E0 ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='27eV at k = 0 in experiments[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' This may not necessarily be true, however, as we consider a perylene filled cavity only to achieve frequency separation of photon modes with different polarization, and this can instead be easily achieved with an electrically tunable liquid crystal cavity [10], replacing a perylene filled cavity with a liquid-crystal cavity will not modify the underlying physics of the phenomenon we are interested in, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=', the idea of using saturation to break TRS will remain intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Porphyrin, Ce:YAG, and monolayer MoS2 We take areal density ρA = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='55 × 105µm−2 (∼ 2000 molecules in 75nm ××× 75nm)[11], relative permittivity ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='5[12], frequency ¯hωe = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='8056eV and transition dipole µ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='1184au × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='5417D/au = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='84D [13] for the porphyrin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Also, we consider 100 such porphyrin films stacked one over the other along the z direction within the cavity to achieve strong light-matter coupling, Nz = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Therefore, the effective areal density of molecules ρ′ A = NzρA will be used instead of ρA while computing Jk,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' These are the parameters used to generate Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Similarly, using density ρYAG = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='11g cm−3, molar mass MYAG = 738 g mol−1, number of Y3+ per unit cell nY3+ = 3, and concentration of Ce3+ (relative to Y3+) 1% = 10−2 [14], we obtain the effective areal density of Ce3+ ions in a L′ z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='1µm thick layer of Ce:YAG to be ρ′ A = 10−2L′ znY3+ρYAGNA/MYAG = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='25 × 107µm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' This will be used while computing Jk,α in place of ρA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' We use relative permittivity ε = 12[15] and frequency ¯hωe = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='53eV (489nm[16]) for the transition in a Ce:YAG crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Using the oscillator strength of this transition 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='286[16], we calculate the transition dipole µ0 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='46D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' These are the parameters used to generate Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' For monolayer MoS2, we consider A-excitons at ¯hωe = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='855eV[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' From Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [17], we take the Rabi splitting at resonance, and use µ0√ρA � ¯hωe/2Lzεε0 ≈ 39meV/2 = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='5meV in our calculations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 5f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' SUPPLEMENTARY REFERENCES [1] Fabre, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' & Treps, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Modes and states in quantum optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Reviews of Modern Physics 92, 035005 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [2] Zoubi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' & La Rocca, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Microscopic theory of anisotropic organic cavity exciton polaritons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Physical Review B 71, 235316 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [3] Martinelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' & Martelli, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Polarization, mirrors, and reciprocity: birefringence and its compensation in optical retracing circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Advances in Optics and Photonics 9, 129–168 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [4] Asb´oth, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=', Oroszl´any, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' & P´alyi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' A short course on topological insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Lecture notes in physics 919, 166 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [5] Silveirinha, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Chern invariants for continuous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Physical Review B 92, 125153 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [6] Fukui, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=', Hatsugai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' & Suzuki, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Chern numbers in discretized brillouin zone: efficient method of computing (spin) hall conductances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Journal of the Physical Society of Japan 74, 1674–1677 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [7] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Ribeiro, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Theory for nonlinear spectroscopy of vibrational polaritons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' The journal of physical chemistry letters 9, 3766–3771 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' 6 [8] Fowler-Wright, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=', Lovett, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' & Keeling, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Efficient many-body non-markovian dynamics of organic polaritons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Physical Review Letters 129, 173001 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [9] Ren, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nontrivial band geometry in an optically active system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nature communications 12, 1–8 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [10] Rechci´nska, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Engineering spin-orbit synthetic hamiltonians in liquid-crystal optical cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Science 366, 727–730 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [11] Hulsken, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Real-time single-molecule imaging of oxidation catalysis at a liquid–solid interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nature nanotechnology 2, 285–289 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [12] Li, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=', Swanson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=', Robinson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' & Hoffbauer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Porphyrin based self-assembled monolayer thin films: synthesis and characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Journal of the American Chemical Society 115, 6975–6980 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [13] Sun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=', Gu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' & Mukamel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Polariton ring currents and circular dichroism of mg-porphyrin in a chiral cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Chemical Science (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [14] Bachmann, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=', Ronda, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' & Meijerink, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Temperature quenching of yellow ce3+ luminescence in yag: Ce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Chemistry of Materials 21, 2077–2084 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [15] Ctibor, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=', Sedl´aˇcek, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' & Hudec, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Dielectric properties of ce-doped yag coatings produced by two techniques of plasma spraying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Bolet´ın de la Sociedad Espa˜nola de Cer´amica y Vidrio (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [16] Kolesov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Mapping spin coherence of a single rare-earth ion in a crystal onto a single photon polarization state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Physical review letters 111, 120502 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' [17] Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=', Cain, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=', Stanev, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=', Dravid, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' & Stern, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Valley-polarized exciton–polaritons in a monolayer semiconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} +page_content=' Nature Photonics 11, 431–435 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQflgTe/content/2301.03287v1.pdf'} diff --git a/39E4T4oBgHgl3EQfAwsS/content/2301.04845v1.pdf b/39E4T4oBgHgl3EQfAwsS/content/2301.04845v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7325c994bb53c6579ef56c69fce7f8cd3190695a --- /dev/null +++ b/39E4T4oBgHgl3EQfAwsS/content/2301.04845v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:58a5652554d08d2a957dd46d54887370eea82d5eefadbd775e0b4e734a93d94e +size 1715202 diff --git a/39E4T4oBgHgl3EQfAwsS/vector_store/index.faiss b/39E4T4oBgHgl3EQfAwsS/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..d4f0145fdabf89dabe6c01fb275da4a71d3ada88 --- /dev/null +++ b/39E4T4oBgHgl3EQfAwsS/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:006b19ad3ed0dbc9cb0853e38a0d3069daeb38c29120d6d831adb14c6b092bad +size 23855149 diff --git a/3NFIT4oBgHgl3EQf5iun/vector_store/index.faiss b/3NFIT4oBgHgl3EQf5iun/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..331686ded603031cface24beed554f096f5754c6 --- /dev/null +++ b/3NFIT4oBgHgl3EQf5iun/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d272f5f3beb37374182c18db09966d6908f2a858a53ef7d14cc626c6155656d9 +size 655405 diff --git a/3NFIT4oBgHgl3EQf5iun/vector_store/index.pkl b/3NFIT4oBgHgl3EQf5iun/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..04e45da94e6df3b92fe848615def947926c1769b --- /dev/null +++ b/3NFIT4oBgHgl3EQf5iun/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:14328f32db04dc9b84c3fd8537ccbc40b62404d33409718f0ddbfa8927ce48cd +size 25620 diff --git a/3dAzT4oBgHgl3EQf9P73/vector_store/index.faiss b/3dAzT4oBgHgl3EQf9P73/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..e720cc4cd1d16c86895d8b08e8d3aadf5492c4a1 --- /dev/null +++ b/3dAzT4oBgHgl3EQf9P73/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:06b376a548424cc9c7c7d3e2a7f983f885a4450274184dd904c216d45cacc11f +size 5767213 diff --git a/3dAzT4oBgHgl3EQf9P73/vector_store/index.pkl b/3dAzT4oBgHgl3EQf9P73/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..4e2664d721cf2acf901f3c637f56a91de22eb67f --- /dev/null +++ b/3dAzT4oBgHgl3EQf9P73/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0e69246507a728b3159de841b94d951b79c8c2dd114751cd50804fe95b9e8237 +size 203408 diff --git a/4NFKT4oBgHgl3EQfRi0P/vector_store/index.faiss b/4NFKT4oBgHgl3EQfRi0P/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..3356eaf8808a33a4b86d09ca00cc623ef7dea8d5 --- /dev/null +++ b/4NFKT4oBgHgl3EQfRi0P/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d724d7c8865138e1bb0bc929b057fe4d22ccaadc5867672f116ca26fffe72416 +size 3538989 diff --git a/4NFKT4oBgHgl3EQfRi0P/vector_store/index.pkl b/4NFKT4oBgHgl3EQfRi0P/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..72b0a8f677d08201fc2bfb3c120d2a1c9d6eaac3 --- /dev/null +++ b/4NFKT4oBgHgl3EQfRi0P/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:def8566d86854ac1086a9d934c0889d8bc1b976945a488a165380f87a9ea7af4 +size 120055 diff --git a/4dAzT4oBgHgl3EQfuv2o/content/tmp_files/2301.01696v1.pdf.txt b/4dAzT4oBgHgl3EQfuv2o/content/tmp_files/2301.01696v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d288c1fa7d71602dca2394f6bbebd54acc651467 --- /dev/null +++ b/4dAzT4oBgHgl3EQfuv2o/content/tmp_files/2301.01696v1.pdf.txt @@ -0,0 +1,2917 @@ +Parameterised and Fine-grained Subgraph +Counting, modulo 2∗ +Leslie Ann Goldberg +University of Oxford +Marc Roth +University of Oxford +Abstract +Given a class of graphs H, the problem ⊕Sub(H) is defined as follows. The input is a graph H ∈ H +together with an arbitrary graph G. The problem is to compute, modulo 2, the number of subgraphs +of G that are isomorphic to H. The goal of this research is to determine for which classes H the +problem ⊕Sub(H) is fixed-parameter tractable (FPT), i.e., solvable in time f(|H|) · |G|O(1). +Curticapean, Dell, and Husfeldt (ESA 2021) conjectured that ⊕Sub(H) is FPT if and only if the +class of allowed patterns H is matching splittable, which means that for some fixed B, every H ∈ H +can be turned into a matching (a graph in which every vertex has degree at most 1) by removing at +most B vertices. +Assuming the randomised Exponential Time Hypothesis, we prove their conjecture for (I) all +hereditary pattern classes H, and (II) all tree pattern classes, i.e., all classes H such that every +H ∈ H is a tree. +We also establish almost tight fine-grained upper and lower bounds for the case of hereditary +patterns (I). +2012 ACM Subject Classification Theory of computation → Problems, reductions and completeness; +Mathematics of computing → Discrete mathematics +Keywords and phrases modular counting, parameterised complexity, fine-grained complexity, sub- +graph counting +Acknowledgements We want to thank Radu Curticapean, Holger Dell and Thore Husfeldt for +insightful discussions on an early draft of this work. +1 +Introduction +The last two decades have seen remarkable progress in the classification of subgraph counting +problems: Given a small pattern graph H and a large host graph G, how often does H occur +as a subgraph if G? Since it was discovered that subgraph counts from small patterns reveal +global properties of complex networks [25, 26], subgraph counting has also found several +applications in fields such as biology [1, 32] genetics [34], phylogeny [24], and data mining [35]. +Moreover, the theoretical study of subgraph counting and related problems has led to many +deep structural insights, establishing both new algorithmic techniques and tight lower bounds +under the lenses of fine-grained and parameterised complexity theory [18, 15, 9, 13, 12, 5, 3]. +Without any additional restrictions, the subgraph counting problem is infeasible. The +complexity class #W[1] is the parameterised complexity class analgous to NP (see Section 2 +for more detail). Under standard assumptions, problems that are #W[1]-hard are not fixed- +parameter tractable (FPT). However, the canonical complete problem for #W[1], the problem +of counting k-cliques, corresponds to the special case of the subgraph counting problem +∗ For the purpose of Open Access, the authors have applied a CC BY public copyright licence to any +Author Accepted Manuscript version arising from this submission. All data is provided in full in the +results section of this paper. +arXiv:2301.01696v1 [cs.CC] 4 Jan 2023 + +2 +Parameterised and Fine-grained Subgraph Counting, modulo 2 +where H is a clique of size k. This problem cannot be solved in time f(k) · no(k) for any +function f unless the Exponential Time Hypothesis (ETH) fails [7, 8]. Due to this hardness +result, the research focus in this area shifted to the question: Under which restrictions on the +patterns H and the hosts G is algorithmic progress possible? More precisely, under which +restrictions can the problem be solved in time f(|H|) · |G|O(1), for some computable function +f? Instances that can be solved within such a run time bound are called fixed-parameter +tractable (FPT); allowing a potential super-polynomial overhead in the size of the pattern +|H| formalises the assumption that H is assumed to be (significantly) smaller than G. +If only the patterns are restricted, then the situation if fully understood. Formally, given +a class H of patterns, the problem #Sub(H) asks, given as input a graph H ∈ H and an +arbitrary graph G, to compute the number of subgraphs of G that are isomorphic to H. +Following initial work by Flum and Grohe [18] and by Curticapean [10], Curticapean and +Marx [13] proved that, under standard assumptions, #Sub(H) is FPT if and only if H has +bounded matching number, that is, if there is a positive integer B such that the size of +any matching in any graph in H is at most B. They also proved that all FPT cases are +polynomial-time solvable. +In stark contrast, almost nothing is known for the decision version Sub(H). Here, the +task is to correctly decide whether there is a copy of H ∈ H in G, rather than to count +the copies. It is known that Sub(H) is FPT whenever H has bounded treewidth (see e.g. +[19, Chapter 13]), and it is conjectured that those are all FPT cases. However, resolving +this conjecture belongs to the “most infamous” open problems in parameterised complexity +theory [17, Chapter 33.1] +1.1 +Counting Modulo 2 +To interpolate between the fully understood realm of (exact) counting and the barely +understood realm of decision, Curticapean, Dell and Husfeldt proposed the study of counting +subgraphs, modulo 2 [11]. Formally, they introduced the problem ⊕Sub(H), which expects +as input a graph H ∈ H and an arbitrary graph G, and the goal is to compute modulo 2 the +number of subgraphs of G isomorphic to H. +The study of counting modulo 2 received significant attention from the viewpoint of +classical and structural complexity theory. For example, one way to state Toda’s Theorem [33] +is PH ⊆ P⊕P, implying that counting satisfying assignments of a CNF, modulo 2, is at least +as hard as the polynomial hierarchy. Another example is the quest to classify the complexity +of counting modulo 2 the homomorphisms to a fixed graph, which was very recently resolved +by Bulatov and Kazeminia [6]. +In their work [11], Curticapean, Dell and Husfeldt proved that the problem of counting +k-matchings modulo 2, that is, the problem ⊕Sub(H) where H is the class of all 1-regular +graphs, is fixed-parameter tractable, where the parameter k is |H|. Since the exact counting +version of this problem is #W[1]-hard [10], their result provides an example where counting +modulo 2 is strictly easier than exact counting (subject to complexity assumptions). The +complexity class ⊕W[1] can be defined via the complete problem of counting k-cliques +modulo 2. Crucially, ⊕W[1]-hard problems are not fixed-parameter tractable, unless the +randomised ETH (rETH) fails. Curticapean et al. [11] proved that counting k-paths modulo +2 is ⊕W[1]-hard. Since finding a k-path in a graph G is fixed-parameter tractable via colour- +coding [2], this hardness result provides an example where counting modulo 2 is strictly +harder than decision (subject to complexity assumptions). Combining those observations, +it appears that counting subgraphs modulo 2 may lie strictly in between the complexity of +decision and the complexity of exact counting. + +L. A. Goldberg and M. Roth +3 +A matching is a graph whose degree is at most 1. The matching-split number of a graph +H is the minimum size of a set S ⊆ V (H) such that H \ S is a matching. A class of graphs +H is called matching splittable if there is a positive integer B such that the matching-split +number of any H ∈ H is at most B. For example, the class of all matchings is matching +splittable while the class of all cycles is not. Curticapean, Dell and Husfeldt extended their +FTP algorithm for counting k-matchings modulo 2 to obtain an FPT algorithm for ⊕Sub(H) +for any matching-splittable class H. On this basis, they then made the following conjecture. +▶ Conjecture 1 ([11]). ⊕Sub(H) is FPT if and only if H is matching splittable. +A class H of graphs is called hereditary if it is closed under vertex removal. Intriguingly, +if Conjecture 1 is true, then the FPT criterion for counting subgraphs modulo 2 (⊕Sub(H)) +would coincide with the polynomial-time criterion for finding subgraphs (Sub(H)) for hered- +itary pattern classes H as established by Jansen and Marx. +▶ Theorem 2 ([23]). Let H be a hereditary class of graphs and assume P ̸= NP. Then +Sub(H) is solvable in polynomial time if and only if H is matching splittable. +Jansen and Marx also conjecture that the condition of H being hereditary can be removed. +▶ Conjecture 3 ([23]). Sub(H) is solvable in polynomial time if and only if H is matching +splittable. +Conjectures 1 and 3 have the remarkable consequence that ⊕Sub(H) is FPT if and only +if Sub(H) is solvable in polynomial time. In the current work we establish this consequence +for all hereditary pattern classes. +1.2 +Our Contributions +We resolve Conjecture 1 for all hereditary classes H, as well as for every class H consisting +only of trees. +▶ Theorem 4. Let H be a hereditary class of graphs. If H is matching splittable, then +⊕Sub(H) is fixed-parameter tractable. +Otherwise, the problem is ⊕W[1]-complete and, +assuming rETH, cannot be solved in time f(|H|) · |G|o(|V (H)|/ log |V (H)|) for any function f. +▶ Theorem 5. Let T be a recursively enumerable class of trees. If T is matching splittable, +then ⊕Sub(T ) is fixed-parameter tractable. Otherwise ⊕Sub(T ) is ⊕W[1]-complete. +In order to prove our classifications, we adapt the by-now-standard technique for ana- +lysing subgraph counting problems established by Curticapean, Dell and Marx [12]. Let +#Sub(H → G) denote the number of subgraphs of a graph G that are isomorphic to a +graph H and let #Hom(F → G) denotes the number of homomorphisms (edge-preserving +mappings) from a graph F to a graph G. Given a graph H, there is a function aH from +graphs to rationals with finite support such that the following holds for any graph G: +#Sub(H → G) = +� +F +aH(F) · #Hom(F → G) , +(1) +where the sum is over all (isomorphism types of) graphs. Since aH has finite support, +aH(F) = 0 for all but finitely-many graphs F. Thus, equation (1) allows us to express the +solution to the exact counting problem as a finite linear combination of homomorphism counts. +In a nutshell, the framework of [12] states that computing the function G �→ #Sub(H → G) + +4 +Parameterised and Fine-grained Subgraph Counting, modulo 2 +is hard to compute if and only if there is a graph F of high treewidth with aH(F) ̸= 0. +This translates the complexity of (exact) subgraph counting to the purely combinatorial +problem of understanding the coefficients aH. One might hope that this strategy transfers +to counting modulo 2 as well. Unfortunately, this is not possible as Equation (1) might +not be well-defined if arithmetic is done modulo 2. The reason for this is the fact that the +coefficients aH(F) are of the form µ(F, H) × |Aut(H)|−1, where µ(F, H) is an integer, and +Aut(H) is the automorphism group of the graph H [12]. Thus there is, a priori, no hope +to extend the framework to counting modulo 2 for pattern graphs with an even number of +automorphisms. In fact, according to Curticapean, Dell and Husfeldt [11], the absence of a +comparable framework for counting modulo 2 is one of the main challenges for establishing +the hardness part of Conjecture 1, and it is the main reason why the reductions in [11] use +more classical, gadget-based reductions. +In this work, we solve the problem of patterns with an even number of automorphisms +by considering a colourful intermediate problem. More concretely, we will equip each edge +of the pattern H with a distinct colour and show that it will be sufficient to consider only +automorphisms that preserve the colours. If H has no isolated vertices, then this is only +the trivial automorphism. Formally, the coloured approach will be based on the notion of +so-called fractured graphs introduced by Peyerimhoff et al. [29]. +In what follows (Section 2), we will first introduced all required notions and previous +results. In Section 3, we will prove the classification for hereditary pattern classes (Theorem 4). +On a technical level, this proof can be considered a warm-up for the significantly harder +challenge of establishing the classification for trees (Theorem 5), which we prove in Section 4. +2 +Preliminaries +Let f : A1 × A2 → B be a function. For each a1 ∈ A1 we write f(a1, ⋆) : A2 → B for the +function that maps a2 ∈ A2 to f(a1, a2). +Graphs in this work are undirected and without self loops. A homomorphism from a +graph H to a graph G is a mapping ϕ from the vertices V (H) of H to the vertices V (G) +of G such that for each edge e = {u, v} ∈ E(H) of H, the image ϕ(e) = {ϕ(u), ϕ(v)} is an +edge of G. A homomorphism is called an embedding if it is injective. We write Hom(H → G) +and Emb(H → G) for the sets of homomorphisms and embeddings, respectively, from H +to G. An embedding ϕ ∈ Emb(H → G) is called an isomorphism if it is bijective and +{u, v} ∈ E(H) ⇔ {ϕ(u), ϕ(v)} ∈ E(G). We say that H and G are isomorphic, denoted by +H ∼= G, if an isomorphism from H to G exists. A graph invariant ι is a function from graphs +to rationals such that ι(H) = ι(G) for each pair of isomorphic graphs H and G. +A subgraph of G is a graph G′ with V (G′) ⊆ V (G) and E(G′) ⊆ E(G). We write +Sub(H → G) for the set of all subgraphs of G that are isomorphic to H. Given a subset of +vertices S ⊆ V (G) of a graph G, we write G[S] for the graph induced by S, that is, G[S] has +vertices S and edges {{u, v} ⊆ S | {u, v} ∈ E(G)}. +We denote by tw(G) the treewidth of the graph G. Since we will rely on treewidth purely +in a black-box manner, we omit the technical definition and refer the reader to [14, Chapter +7]. +Given any graph invariant ι (such as treewidth) and a class of graphs G, we say that +ι is bounded in G if there is a non-negative integer B such that, for all G ∈ G, ι(G) ≤ B. +Otherwise we say that ι is unbounded in G. +Given a graph H = (V, E), a splitting set of H is a subset of vertices S such that every +vertex in H[V \S] has degree at most 1. The matching-split number of H is the minimum size + +L. A. Goldberg and M. Roth +5 +v +vB1 vB2 +Figure 1 Illustration of the construction of a fractured graph from [29]. The left picture shows +a vertex v of a graph Q with incident edges EQ(v) = { , , , , , }. The right picture shows the +splitting of v in the construction of the fractured graph Q +♯ +σ for a fracture σ satisfying that the +partition σv contains two blocks B1 = { , , }, and B2 = { , , }. +of a splitting set of H. A class of graphs H is called matching splittable the matching-split +number of H is bounded. +2.1 +Colour-Preserving Homomorphisms and Embeddings +A homomorphism c from a graph G to a graph Q is sometimes called a “Q-colouring” of G. +A Q-coloured graph is a pair consisting of a graph G and a homomorphism c from G to Q. +Note that the identity function idQ on V (Q) is a Q-colouring of Q. If a homomorphism c +from G to Q is vertex surjective, then we call (G, c) a surjectively Q-coloured graph. +▶ Definition 6 (cE). A Q-colouring c of a graph G induces a (not necessarily proper) +edge-colouring cE : E(G) → E(Q) given by cE({u, v}) = {c(u), c(v)}. +Notation: Given a Q-coloured graph (G, c) and a vertex u ∈ V (Q), we will use the +capitalised letter U to denote the subset of vertices of G that are coloured by c with u, that +is, U := c−1(u) ⊆ V (G). +Given two Q-coloured graphs (H, cH) and (G, cG), we call a homomorphism ϕ from H +to G colour-preserving if for each v ∈ V (H) we have cG(ϕ(v)) = cH(v). We note the +special case in which Q = H and cH is the identity idQ; then the condition simplifies to +cG(ϕ(v)) = v. A colour-preserving embedding of (H, cH) in (G, cG) is a vertex injective colour- +preserving homomorphism from (H, cH) to (G, cG). We write Hom((H, cH) → (G, cG)) and +Emb((H, cH) → (G, cG)) for the sets of all colour-preserving homomorphisms and embeddings, +respectively, from (H, cH) to (G, cG). +Let k be a positive integer, let H be a graph with k edges, and let (G, γ) be a pair +consisting of a graph G and a function that maps each edge of G to one of k distinct colours. +We refer to γ as a “k-edge colouring” of G. For example, in most of our applications we will fix +a graph Q with k edges and a Q-colouring c of G and we will take γ to be the edge-colouring +cE from Definition 6. We write ColSub(H → (G, γ)) for the set of all subgraphs of G that +are isomorphic to H and that contain each of the k edge colours precisely once. +2.2 +Fractures and Fractured Graphs +In this work, we will crucially rely on and extend the framework of fractured graphs as +introduced in [29]. +▶ Definition 7 (Fractures). Let Q be a graph. For each vertex v of Q, let EQ(v) be the set +of edges of Q that are incident to v. A fracture of Q is a tuple ρ = (ρv)v∈V (Q), where for +each vertex v of Q, ρv is a partition of EQ(v). + +6 +Parameterised and Fine-grained Subgraph Counting, modulo 2 +Note that a fracture describes how to split (or how to fracture) each vertex of a given +graph: for each vertex v, create a vertex vB for each block B in the partition ρv; edges +originally incident to v are made incident to vB if and only if they are contained in B. We +call the resulting graph the fractured graph H +♯ +ρ; a formal definition is given in Definition 8, +a visualisation is given in Figure 1. +▶ Definition 8 (Fractured Graph Q +♯ +ρ). Given a graph Q, we consider the matching MQ +containing one edge for each edge of Q; formally, +V (MQ) := +� +e={u,v}∈E(Q) +{ue, ve} +and +E(MQ) := {{ue, ve} | e = {u, v} ∈ E(Q)}. +For a fracture ρ of Q, we define the graph Q +♯ +ρ to be the quotient graph of MQ under +the equivalence relation on V (MQ) which identifies two vertices ve, wf of MQ if and only if +v = w and e, f are in the same block B of the partition ρv of EQ(v). We write vB for the +vertex of Q +♯ +ρ given by the equivalence class of the vertices ve (for which e ∈ B) of MQ. +▶ Definition 9 (Canonical Q-colouring cρ). Let Q be a graph and let ρ be a fracture of Q. +The canonical Q-colouring of the fractured graph Q +♯ +ρ maps vB to v for each v ∈ V (Q) and +block B ∈ ρv, and is denoted by cρ. +Observe that cρ is the identity in V (Q) if ρ is the coarsest fracture (that is, each partition +ρv only contains one block, in which case Q +♯ +ρ = Q). +2.3 +Parameterised and Fine-grained Computation +A parameterised computational problem is a pair consisting of a function P : Σ∗ → {0, 1} and +a computable parameterisation κ : Σ∗ → N. A fixed-parameter tractable (FPT) algorithm for +(P, κ) is an algorithm that computes P and runs, on input x ∈ Σ∗, in time f(κ(x)) · |x|O(1) +for some computable function f. We call (P, κ) fixed-parameter tractable (FPT) if an FPT +algorithm for (P, κ) exists. +A parameterised Turing-reduction from (P, κ) to (P ′, κ′) is an FPT algorithm for (P, κ) +that is equipped with oracle access to P ′ and for which there is a computable function g such +that, on input x, each oracle query y satisfies κ′(y) ≤ g(κ(x)). We write (P, κ) ≤fpt +T (P ′, κ′) +if a parameterised Turing-reduction from (P, κ) to (P ′, κ′) exists. This guarantees that +fixed-parameter tractability of (P ′, κ′) implies fixed-parameter tractability of (P, κ). For a +more comprehensive introduction, we refer the reader the standard textbooks [14] and [19]. +Counting modulo 2 and the rETH +The lower bounds in this work will rely on the hardness of the parameterised complexity +class ⊕W[1], which can be considered a parameterised equivalent of ⊕P. Following [11], we +define ⊕W[1] via the complete problem ⊕Clique: Given as input a graph G and a positive +integer k, the goal is to compute the number of k-cliques in G modulo 2, i.e., to compute +⊕Sub(Kk → G). The problem is parameterised by k. A parameterised problem (P, κ) is +called ⊕W[1]-hard if ⊕Clique ≤fpt +T (P, κ), and it is called ⊕W[1]-complete if, additionally, +(P, κ) ≤fpt +T ⊕Clique. +Modifications of the classical Isolation Lemma (see e.g. [4] and [36]) yield a randomised +parameterised Turing reduction from finding a k-clique to computing the parity of the +number of k-cliques. In combination with existing fine-grained lower bounds for finding a +k-clique [7, 8], it can then be shown that ⊕Clique cannot be solved in time f(k) · |G|o(k) +for any function f, unless the randomised Exponential Time Hypothesis fails: + +L. A. Goldberg and M. Roth +7 +▶ Definition 10 (rETH, [22]). The randomised Exponential Time Hypothesis (rETH) asserts +that 3-SAT cannot be solved by a randomised algorithm in time exp o(n), where n is the +number of variables of the input formula. +As an immediate consequence, the rETH implies that ⊕W[1]-hard problems are not fixed- +parameter tractable. +For the lower bounds in this work, we won’t reduce from ⊕Clique directly, but instead +from the following, more general problem: +▶ Definition 11 (⊕cp-Hom). Let H be a class of graphs. The problem ⊕cp-Hom(H) has +as input a graph H ∈ H and a surjectively H-coloured graph (G, c). The goal is to compute +⊕Hom((H, idH) → (G, c)). The problem is parameterised by |H|. +The following lower bound was proved independently in [27, 29] and [11]. +▶ Theorem 12. Let H be a recursively enumerable class of graphs. If the treewidth of H is +unbounded then ⊕cp-Hom(H) is ⊕W[1]-hard and, assuming the rETH, it cannot be solved +in time f(|H|) · |G|o(tw(H)/ log tw(H)) for any function f. +Next is the central problem in this work. +▶ Definition 13 (⊕Sub). Let H be a class of graphs. The problem ⊕Sub(H) has as input +a graph H ∈ H and a graph G. The goal is to compute ⊕Sub(H → G). The problem is +parameterised by |H|. +For example, writing K for the set of all complete graphs, the problem ⊕Sub(K) is +equivalent to ⊕Clique. +Complexity Monotonicity and Inclusion-Exclusion +Throughout this work, we will rely on two important tools introduced in [29]. For the sake +of being self-contained, we encapsulate them below in individual lemmas. +The first tool is an adaptation of the so-called Complexity Monotonicity principle to +the realm of fractured graphs and modular counting (see [29, Sections 4.1 and 6.3] for a +detailed treatment and for a proof). Intuitively, the subsequent lemma states that evaluating, +modulo 2, a linear combination of colour-prescribed homomorphism counts from fractured +graphs, is as hard as evaluating its hardest term with an odd coefficient. +▶ Lemma 14 ([29]). There is a deterministic algorithm A and a computable function f such +that the following conditions are satisfied: +1. A expects as input a graph Q and a Q-coloured graph (G, c). +2. A is equipped with oracle access to a function +(G′, c′) �→ +� +ρ +a(ρ) · ⊕Hom((Q +♯ +ρ, cρ) → (G′, c′)) +mod 2 , +where the sum is over all fractures of Q and a is a function from fractures of Q to integers. +3. Each oracle query (G′, c′) is of size at most f(|Q|) · |G|. +4. A computes ⊕Hom((Q +♯ +ρ, cρ) → (G, c)) for each fracture ρ with a(ρ) ̸= 0 mod 2. +5. The running time of A is bounded by f(|Q|) · |G|O(1). +The second tool is a standard application of the inclusion-exclusion principle (see e.g. [29, +Sections 4.2 and 6.3]). It will be used in the final steps of our reductions to remove the +colourings. + +8 +Parameterised and Fine-grained Subgraph Counting, modulo 2 +▶ Lemma 15 ([29]). There is a deterministic algorithm A that satisfies the following condi- +tions: +1. A expects as input a graph H with k edges, a graph G and a k-edge colouring γ of G. +2. A is equipped with oracle access to the function ⊕Sub(H → ⋆), and each oracle query G′ +satisfies |G′| ≤ |G|. +3. A computes ⊕ColSub(H → (G, γ)). +4. The running time of A is bounded by 2|H| · |G|O(1). +3 +Classification for Hereditary Graph Classes +In this section, we will completely classify the complexity of ⊕Sub(H) for hereditary classes. +Let us start by restating the classification theorem. +▶ Theorem 4. Let H be a hereditary class of graphs. If H is matching splittable, then +⊕Sub(H) is fixed-parameter tractable. +Otherwise, the problem is ⊕W[1]-complete and, +assuming rETH, cannot be solved in time f(|H|) · |G|o(|V (H)|/ log |V (H)|) for any function f. +The proof of Theorem 4 is split in four cases, which stem from a structural property of +non matching splittable hereditary graph classes H due to Jansen and Marx [23]. For the +statement, we need to consider the following classes: +Fω is the class of all complete graphs. +Fβ is the class of all complete bipartite graphs. +FP2 is the class of all P2-packings, that is, disjoint unions of paths with two edges.1 +FK3 is the class of all triangle packings, that is, disjoint unions of the complete graph of +size 3. +▶ Theorem 16 (Theorem 3.5 in [23]). Let H be a hereditary class of graphs. If H is not +matching splittable then at least one of the following are true: (1.) Fω ⊆ H, (2.) Fβ ⊆ H, +(3.) FP2 ⊆ H, or (4.) FK3 ⊆ H. +Thus, it suffices to consider cases 1. - 4. to prove Theorem 4. We start with the easy +cases of cliques and bicliques; they follow implicitly from previous works [11, 16, 27] and we +only include a proof for completeness. Note that a tight bound under rETH is known for +those cases: +▶ Lemma 17. Let H be a hereditary class of graphs. If Fω ⊆ H or Fβ ⊆ H then ⊕Sub(H) +is ⊕W[1]-hard and, assuming rETH, cannot be solved in time f(|H|) · |G|o(|V (H)|) for any +function f. +Proof. If Fω ⊆ H then ⊕W[1]-hardness follows immediately from the fact that ⊕Clique +is the canonical ⊕W[1]-complete problem [11]. For the rETH lower bound, we can reduce +from the problem of deciding the existence of a k-clique via a (randomised) reduction using a +version of the Isolation Lemma due to Williams et al. [36, Lemma 2.1]. This reduction does +not increase k or the size of the host graph and is thus tight with respect to the well-known +lower bound for the clique problem due to Chen et al. [7, 8]: Deciding the existence of a +k-clique in an n-vertex graph cannot be done in time f(k) · no(k) for any function f, unless +ETH fails. Our lower bound under rETH follows since the reduction is randomised. +1 To avoid confusion, we remark that [23] uses P3 to denote the path of two edges (and three vertices). +In the current work, it will be more convenient to use the number of edges of a path as index. + +L. A. Goldberg and M. Roth +9 +If Fβ ⊆ H, then the claim holds by [16, Theorem 5], which established the problem of +counting, modulo 2, the induced copies of a k-by-k-biclique in an n-vertex bipartite graph +to be ⊕W[1]-hard and not solvable in time f(k) · no(k) for any function f, unless rETH +fails. Since a copy of a biclique (with at least one edge) in a bipartite graph must always be +induced, the claim follows. This concludes the proof of Lemma 17. +◀ +The more interesting cases are FP2 ⊆ H and FK3 ⊆ H. One reason for this is that, in +contrast to cliques and bicliques, the decision version of those instances are fixed-parameter +tractable. Hence a reduction from the decision version via e.g. an isolation lemma does not +help. In other words, establishing hardness for those cases requires us to rely on the full +power of counting modulo 2. More precisely, we will rely on the framework of fractures +graphs (see Section 2). Both cases can be considered simpler applications of the machinery +used in the later sections, so we will present all steps in great detail. While this might seem +unnecessary given the simplicity of the constructions, we hope that it enables the reader to +make themselves familiar with the general reduction strategies which will be used throughout +the later sections of this work. +3.1 +Triangle Packings +The goal of this subsection is to establish hardness of ⊕Sub(FK3). To this end, let ∆ be an +infinite computable class of cubic bipartite expander graphs, and let Q = {L(H) | H ∈ ∆} +where L(H) is constructed as follows: Each v ∈ V (H) becomes a triangle with vertices vx, +vy, and vz corresponding to the three neighbours x, y, and z of v. Finally, for every edge +{u, v} ∈ E(H) we identify vu and uv. In fact, L(H) is just the line graph of H: Every edge of +H becomes a vertex in L(H), and two vertices of L(H) are made adjacent if and only if the +corresponding edges in H are incident. Since all H ∈ ∆ are bipartite (and thus triangle-free), +we can easily observe the following.2 +▶ Observation 18. The mapping v �→ (vx, vy, vz) is a bijection from vertices of H to triangles +in L(H). +We also consider the fracture of L(H) that splits L(H) back into |V (H)| triangles; consider +Figure 2 for an illustration. +▶ Definition 19 (τ(H)). Let H ∈ ∆ and recall that each vertex w of L(H) is obtained by +identifying vu and uv for some edge {u, v} ∈ E(H). Moreover, w has four incident edges +ex, ey, ea, eb, to vx, vy, ua, ub, respectively, where x, y, u are the neighbours of v in H and +v, a, b are the neighbours of u in H. We define τ(H)w := {{ex, ey}, {ea, eb}}, and we proceed +similar for all vertices of L(H). +Next, we use that tw(L(H)) = Ω(tw(H)) (see e.g. [21]). Moreover, tw(L(H)) ≤ |V (L(H))| +since the treewidth of a graph is always bounded by the number of its vertices. Additionally, +|V (L(H))| = |E(H)| by construction. Since the graphs in ∆ are cubic, we further have that +|E(H)| = Θ(|V (H)|) for H ∈ ∆. We combine those bounds with the fact that expander +graphs have treewidth linear in the number of vertices (see e.g. [20]); therefore ∆ and thus +Q have unbounded treewidth. Putting these facts together, we obtain the following. +▶ Fact 20. Q has unbounded treewidth and tw(L(H)) = Θ(|V (L(H))|) = Θ(|V (H)|) for +H ∈ ∆. +2 Observation 18 is also an immediate consequence of Whitney’s Isomorphism Theorem implying that a +triangle of a line graph corresponds to either a claw or to a triangle in its primal graph. + +10 +Parameterised and Fine-grained Subgraph Counting, modulo 2 +We are now able to establish hardness of ⊕Sub(FK3). The proof will heavily rely on the +transformation from edge-coloured subgraphs to homomorphisms established in [29]. +▶ Lemma 21. The problem ⊕Sub(FK3) is ⊕W[1]-hard. Furthermore, on input kK3 and G, +the problem cannot be solved in time f(k) · |G|o(k/ log k) for any function f, unless rETH fails. +Proof. We reduce from ⊕cp-Hom(Q), which, by Fact 20 and Theorem 12, is ⊕W[1]-hard +and for L(H) ∈ Q, it cannot be solved in time f(|L(H)|) · |G|o(|V (L(H))|/ log |V (L(H))|), unless +rETH fails. +Let L and (G, c) be an input instance to ⊕cp-Hom(Q). Recall that ∆ is computable — +that is, there is an algorithm that takes a graph H and determines whether it is in ∆. Thus, +there is an algorithm that takes input L ∈ Q and finds a graph H ∈ ∆ with L = L(H). The +run time of this algorithm depends on |L| but clearly not on (G, c). Let k = |V (H)| and +note that |E(L(H))| = 3k, since, by construction, each vertex v of H becomes a triangle of +L(H). We consider the graph G as a 3k-edge-coloured graph, coloured by cE. That is, each +edge e = {x, y} of G is assigned the colour cE(e) = {c(x), c(y)} which is an edge of L (see +Figure 2 for an illustration). +Now, for any L-coloured graph (G′, c′) recall that ColSub(kK3 → (G′, c′ +E)) is the set of +subgraphs of G′ that are isomorphic to kK3 and that include each edge colour (each edge of +L) precisely once. We will see later that ⊕ColSub(kK3 → (G′, c′ +E)) can be computed using +our oracle for ⊕Sub(FK3) using the principle of inclusion and exclusion. +It was shown in [29, Lemma 4.1] that there is a unique function a such that for every +L-coloured graph (G′, c′) we have3 +#ColSub(kK3 → (G′, c′ +E)) = +� +ρ +a(ρ) · Hom(L +♯ +ρ → (G′, c′)) . +(2) +where the sum is over all fractures of L. Additionally, it was shown in [29, Corollary 4.3] +that +a(⊤) = +� +ρ∈F(kK3,L) +� +w∈V (L) +(−1)|ρw|−1 · (|ρw| − 1)! , +(3) +where ⊤ is the fracture in which each partition consists only of one block (that is, L +♯ +⊤ = L), +and F(kK3, L) is the set of all fractures ρ of L such that L +♯ +ρ ∼= kK3. However, note that, +by Observation 18, there is only way to fracture L into k disjoint triangles, and this fracture +is given by τ(H). Thus, (3) simplifies to +a(⊤) = +� +w∈V (L) +(−1)|τ(H)w|−1 · (|τ(H)w| − 1)! , +(4) +which is odd since each partition of τ(H) consists of precisely two blocks (so in fact the +expression in (4) is (−1)|V (L)|). +Note that the algorithm for ⊕cp-Hom(Q) is supposed to compute ⊕Hom((L, idL) → (G, c)) +which is equal to ⊕Hom(L +♯ +⊤ → (G, c⊤)). Since a(⊤) is odd, we can invoke Lemma 14 to +recover this term by evaluating the entire linear combination (2), that is, by evaluating +the function ⊕ColSub(kK3 → ⋆). More concretely, this means that we need to compute +⊕ColSub(kK3 → (G′, c′ +E)) for some L-coloured graphs (G′, c′) of size at most f(|L|) · |G| for +3 In the language of [29], Equation (2) is obtained by choosing Φ as the property of being isomorphic +to kK3. + +L. A. Goldberg and M. Roth +11 +Figure 2 (Top:) A cubic bipartite graph H ∈ ∆, its line graph L(H), and the fractured graph +induced by τ(H). (Below:) An L(H)-coloured graph (G, c); emphasised in distinct colours is the +edge-colouring cE of G induced by the mapping {u, v} �→ {c(u), c(v)}. Additionally we depict an +element S ∈ ColSub(kK3 → (G, cE)), that is, a subgraph of G isomorphic to kK3 that contains each +edge colour of G precisely once. + +12 +Parameterised and Fine-grained Subgraph Counting, modulo 2 +some computable function f (see 3. in Lemma 14). This can easily be done using Lemma 15 +since we have oracle access to the function ⊕Sub(kK3 → ⋆). We emphasise that, by condition +2. of Lemma 15, each oracle query ˆG satisfies | ˆG| ≤ |G′|, where (G′, c′) is the L-coloured +graph for which we wish to compute ⊕ColSub(kK3 → (G′, c′ +E)). Since |(G′, c′)| ≤ f(|L|) · |G|, +we obtain that | ˆG| ≤ f(|L|) · |G| as well. +Since, by Fact 20, k = Θ(|kK3|) = Θ(|V (L)|) = Θ(tw(L)), our reduction yields ⊕W[1]- +hardness and transfers the conditional lower bound under rETH as desired. +◀ +3.2 +P2-packings +Next we establish hardness for the case of P2-packings. The strategy will be similar in spirit +to the construction for triangle packings; however, rather then identifying a unique fracture +for which the technique applies, we will encounter an odd number of possible fractures in the +current section. +Let ∆ be a computable infinite class of 4-regular expander graphs, and let Q be the class +of all subdivisions of graphs in ∆, that is Q = {H2 | H ∈ ∆}, where H2 is obtained from H +by subdividing each edge once. +We start by establishing an easy but convenient fact on the treewidth of the graphs in Q. +▶ Lemma 22. Q has unbounded treewidth and tw(H2) = Θ(|V (H)|) for H ∈ ∆. +Proof. As in Section 3.1, tw(H) = Θ(|V (H)|) for H ∈ ∆, since expanders have treewidth +linear in the number of vertices. Since H is a minor of H2, and since taking minors cannot +increase treewidth (see [14, Exercise 7.7]), we thus have that tw(H2) = Ω(|V (H)|)). Finally, +we have tw(H2) ≤ |V (H2)| since the treewidth is at most the number of vertices, and +|V (H2)| = O(|V (H)|) since H is 4-regular. In combination, we obtain tw(H2) = Θ(|V (H)|) +for H ∈ ∆. Note that this also implies that Q has unbounded treewidth (as ∆ is infinite). +◀ +For what follows, given a subdivision H2 of a graph H, it will be convenient to assume +that V (H2) = V (H) ∪ SE, where SE = {se | e ∈ E(H}) is the set of the subdivision vertices. +▶ Definition 23 (Odd Fractures). Let H ∈ ∆ and let τ be a fracture of H2. We say that τ is +odd if the following two conditions are satisfied: +1. For each s ∈ SE the partition τs consists of two singleton blocks. +2. For each v ∈ V (H) the partition τv consists of two blocks of size 2. +Consider Figure 3 for a depiction of an odd fracture. +The following two lemmas are crucial for our construction. +▶ Lemma 24. Let H ∈ ∆. The number of odd fractures of H2 is odd. +Proof. The first condition in Definition 23 leaves only one choice for subdivision vertices. +Let us thus consider a vertex v ∈ V (H) = V (H2) \ SE. Since H is 4-regular, there are 4 +incident edges to v. Now note that there are precisely 3 partitions of a 4-element set with two +blocks of size 2. Thus the total number of odd fractures of H2 is 3|V (H)|, which is odd. +◀ +▶ Lemma 25. Let H ∈ ∆, let k = 2|V (H)| and let τ be a fracture of H2 such that τv consists +of at most 2 blocks for each v ∈ V (H2). Then H2 +♯ +τ ∼= kP2 if and only if τ is odd. +Proof. First observe that |E(H2)| = 2|E(H)| = 4|V (H)| = 2k. Thus the number of edges of +H2 +♯ +τ is equal to 2k (for each fracture τ of H2), which is also equal to the number of edges +of kP2. + +L. A. Goldberg and M. Roth +13 +Figure 3 (Top:) Subdividing a 4-regular expander in ∆ depicted by the neighbourhood of an +individual vertex. (Centre:) Illustrations of odd fractures (Definition 23). For each non-subdivision +vertex, there are only three ways to satisfy 2. in Definition 23. This observation is used in Lemma 24 to +show that the number of odd fractures is a power of 3. (Bottom:) Elements of ColSub(kP2 → (G, cE)) +inducing fractures of H2 such that each partition has at most two blocks. Lemma 25 shows that +those are precisely the odd fractures of H2. + +14 +Parameterised and Fine-grained Subgraph Counting, modulo 2 +Thus, H2 +♯ +τ is isomorphic to kP2 if and only if each connected component of H2 +♯ +τ is +a path of length 2. It follows immediately by Definition 23 that τ being odd implies that +H2 +♯ +τ consists only of disjoint P2. It thus remains to show the other direction. +Assume for contradiction that there is a subdivision vertex s ∈ SE of H2 such that τs +consists of only one block (recall that s has degree 2, thus τs either consists of two singleton +blocks, or of one block of size 2). Let e = {u, v} ∈ E(H) be the edge corresponding to s, that +is, s was created by subdividing e. Since H2 +♯ +τ is a union of P2, we can infer that τv and τu +contain a singleton block (otherwise we would have created a connected component which is +not isomorphic to P2). Now recall that both u and v have degree 4, since H is 4-regular. We +obtain a contradiction as follows: By assumption of the lemma, we know that τv and τu can +have at most two blocks. Since we have just shown that both contain a singleton block, it +follows that both τv and τu contain one further block of size 3. However, a block of size 3 +yields a vertex of degree 3 in the fractured graph H2 +♯ +τ, contradicting the fact that H2 +♯ +τ +consists only of disjoint P2. +Thus we have established that, for each s ∈ SE, the partition τs consists of two singleton +blocks. Given this fact, the only way for H2 +♯ +τ being a disjoint union of P2 is that each +partition τv, for v ∈ V (H) = V (H2) \ SE, consists of two blocks of size 2. +◀ +We are now able to prove our hardness result. +▶ Lemma 26. The problem ⊕Sub(FP2) is ⊕W[1]-hard. Furthermore, on input kP2 and G, +the problem cannot be solved in time f(k) · |G|o(k/ log k) for any function f, unless rETH fails. +Proof. We reduce from ⊕cp-Hom(Q), which, by Lemma 22 and Theorem 12, is ⊕W[1]-hard +and for H′ ∈ Q, it cannot be solved in time f(|H′|) · |G|o(|V (H′)|/ log |V (H′)|), unless rETH +fails. +Let H′ and (G, c) be an input instance to ⊕cp-Hom(Q). There is an algorithm that +takes as input a graph H′ ∈ Q and finds a graph H ∈ ∆ with H′ = H2 — this is basically +2-colouring. The run time of this algorithm depends on |H′| but clearly not on (G, c). Let +k = 2|V (H)| and note that |E(H2)| = 2|E(H)| = 4|V (H)| = 2k. We consider the graph G +as a 2k-edge-coloured graph, coloured by cE. That is, each edge e = {x, y} of G is assigned +the colour cE(e) = {c(x), c(y)} which is an edge of H′ = H2. +Now, for any H2-coloured graph (G′, c′) recall that ColSub(kP2 → (G′, c′ +E)) is the set of +subgraphs of G′ that are isomorphic to kP2 and that include each edge colour (each edge of +H2) precisely once. We will see later that ⊕ColSub(kP2 → (G′, c′ +E)) can be computed using +our oracle for ⊕Sub(FP2) using the principle of inclusion and exclusion. +It was shown in [29, Lemma 4.1] that there is a unique function a such that, for every +H2-coloured graph (G′, c′), +#ColSub(kP2 → (G′, c′ +E)) = +� +ρ +a(ρ) · Hom(H2 +♯ +ρ → (G′, c′)) . +(5) +where the sum is over all fractures of H2. As in Section 3.1 from [29, Corollary 4.3] we know +that +a(⊤) = +� +ρ∈F(kP2,H2) +� +w∈V (H2) +(−1)|ρw|−1 · (|ρw| − 1)! , +(6) +where ⊤ is the fracture in which each partition consists only of one block and F(kP2, H2) is +the set of all fractures ρ of H2 such that H2 +♯ +ρ ∼= kP2. +Our next goal is to show that a(⊤) = 1 mod 2. First, suppose that a fracture ρ contains +a partition ρw with at least three blocks. Then (|ρw| − 1)! = 0 mod 2. Thus such fractures + +L. A. Goldberg and M. Roth +15 +do not contribute to a(⊤) if arithmetic is done modulo 2. Next, note that if, for each w, the +partition ρw contains at most 2 blocks, then +� +w∈V (H2) +(−1)|ρw|−1 · (|ρw| − 1)! = 1 +mod 2. +Let Odd(kP2, H2) be the set of all fractures ρ of H2 such that H2 +♯ +ρ ∼= kP2 and each +partition of ρ consists of at most 2 blocks. Our analysis then yields a(⊤) = |Odd(kP2, H2)| +mod 2. Finally, Lemma 25 states that Odd(kP2, H2) is precisely the set of odd fractures, and +Lemma 24 thus implies that |Odd(kP2, H2)| = 1 mod 2. Consequently, a(⊤) = 1 mod 2 as +well, and we have achieved the goal. +Next we can proceed similarly to the case of triangle packings. As in that case, the goal +is to compute ⊕Hom((H2, idH2) → (G, c))) which is equal to ⊕Hom((H2 +♯ +⊤, c⊤) → (G, c)). +Since a(⊤) is odd, we can invoke Lemma 14 to recover this term by evaluating the entire +linear combination (5), that is, if we can evaluate the function ⊕ColSub(kP2 → ⋆). This can +be done by using Lemma 15. Each call to the oracle is of the form ⊕Sub(kP2 → ˆG) where +| ˆG| is bounded by f(k) · |G|. +Now recall that k ∈ Θ(|V (H)|). By Lemma 22, we thus have k = Θ(tw(H2)). Hence our +reduction yields ⊕W[1]-hardness and transfers the conditional lower bound under rETH as +desired. +◀ +We can now conclude the treatment of hereditary pattern classes by proving Theorem 4, +which we restate for convenience. +▶ Theorem 4. Let H be a hereditary class of graphs. If H is matching splittable, then +⊕Sub(H) is fixed-parameter tractable. +Otherwise, the problem is ⊕W[1]-complete and, +assuming rETH, cannot be solved in time f(|H|) · |G|o(|V (H)|/ log |V (H)|) for any function f. +Proof. The fixed-parameter tractability result was shown in [11]. For the hardness result, +using the fact that H is not matching splittable and Theorem 16 we obtain four cases. +If H contains all cliques or all bicliques, then hardness follows from Lemma 17. +If H contains all triangle packings, then hardness follows from Lemma 21. +If H contains all P2-packings, then hardness follows from Lemma 26. +Since the case distinction is exhaustive, the proof is concluded. +◀ +4 +Classification for Trees +Our overall goal is to prove Theorem 5, which we restate for convenience: +▶ Theorem 5. Let T be a recursively enumerable class of trees. If T is matching splittable, +then ⊕Sub(T ) is fixed-parameter tractable. Otherwise ⊕Sub(T ) is ⊕W[1]-complete. +We start by introducing some terminology for trees which will be used in the remainder +of this section. +▶ Definition 27 (2-paths). A 2-path of length a of a tree T is a path x0, x1, . . . , xa such that +deg(x0) ̸= 2, deg(x1) = · · · = deg(xa−1) = 2 and deg(xa) ̸= 2. +Next we introduce rays, which are restricted 2-paths that will be crucial in our analysis. + +16 +Parameterised and Fine-grained Subgraph Counting, modulo 2 +▶ Definition 28 (source, ray, degL,a, degL, and degNL). Let T be a tree. A source of T is any +vertex with degree greater than 2. A ray of length a of T is a 2-path x0, x1, . . . , xa such that +deg(x0) > 2 and deg(xa) = 1. We call x0 the source of the ray. Given a vertex s of degree +at least 3, we write degL,a(s) for the number of rays of length a with source s. We set +degL(s) := +� +a +degL,a(s) . +Finally, we set degNL(s) := deg(s) − degL(s). +Next, we introduce parameters Fa,b, Sc and Cd. Our goal is then to show that, for every +non-matching-splittable class of trees, at least one of those two parameters is unbounded. +▶ Definition 29 (Forks and Fa,b). Let a, b be positive integers. A source s of a tree T is +called an a-b-fork if degNL(s) = 1 and one of the following is true +a ̸= b and degL,a(s), degL,b(s) > 0. +a = b and degL,a(s) > 1. +The a-b-fork number of T, denoted by Fa,b(T) is the maximum size of an independent set +containing only a-b-forks. Finally, we say that a class of trees T has unbounded fork number +if for every positive integer B there are positive integers a and b and a tree T ∈ T such that +Fa,b(T) ≥ B. +▶ Definition 30 (Stars and Sc). A star of size k > 1 in a tree T is a collection of k distinct +rays that have a common source s. For a positive integer c ≥ 3, a c-star of size k in a tree T +is a collection of k distinct rays of length c that have a common source s. +The c-star number of a tree T, denoted by Sc(T) is the maximum size of a c-star in +T. Finally, we say that a class of trees T has unbounded star number if for every positive +integer B there exists c ≥ 3, and a tree T ∈ T such that Sc(T) ≥ B. +▶ Definition 31 (C-gadgets and Cd). A C-gadget4 of order d and length k in a tree T is a +path x0, . . . , xk such that one of the following is true for each inner vertex xi ∈ {1, . . . , k −1}: +(i) +deg(xi) = 2, that is N(xi) = {xi−1, xi+1}, or +(ii) +xi is a source and every neighbour v ∈ N(xi)\{xi−1, xi+1} is contained in a ray of length +at most d from xi to a leaf. +The Cd-number of a tree T, denoted by Cd(T) is the length of the longest C-gadget of order d. +Finally, we say that a class of trees T has unbounded C-number if there exists d > 0 such +that for every positive integer B, and a tree T ∈ T such that Cd(T) ≥ B. +Note that the ordering of the quantifiers in the definition of the Cd-number is different from +the ordering in the definition of the c-star-number. This is due to technical reasons which +are important for the proof of Lemma 32. +▶ Lemma 32. Let T be a class of trees. If T is not matching splittable, then T has either +unbounded fork number, unbounded star number, or unbounded C-number. +Proof. We can assume that there is an overall bound d on the length of 2-paths in trees in +T : Otherwise, T already has unbounded C-number (see (i) in Definition 31)). Hence the +length of every ray in any tree in T is bounded by d as well. Thus +T has unbounded fork number if and only if for every positive integer B there are +a, b ∈ {1, . . . , d} and a tree T ∈ T such that Fa,b(T) ≥ B. +4 C stands for caterpillar, the shape of which resembles the structure of a C-gadget. + +L. A. Goldberg and M. Roth +17 +T has unbounded C-number if and only if Cd is unbounded in T (see Definition 31)). +T has unbounded star number if and only if for every positive integer s there is a +c ∈ {3, . . . , d} and a tree T ∈ T such that Sc(T) ≥ s. +We split the proof into two cases. +Case 1. T has unbounded diameter. +In Case 1, we show that T has unbounded fork number or unbounded C-number. If Cd +is unbounded in T then T has unbounded C-number and we are done so assume that there +is a constant h such that Cd(T) ≤ h for every T ∈ T . +Now let B be a positive integer. We show that there are a, b ∈ {1, . . . , d} and T ∈ T +with Fa,b(T) ≥ B. To this end, we use the premise that T has unbounded diameter. Let +k > (h + 2)(Bd2 + 1) be a positive integer, and let T ∈ T be such that there is a path +P = s, p0, . . . , pk, t in T. Observe that the deletion of all edges in P decomposes T into a +family of disjoint subtrees. We write Ti for the subtree that contains pi. Now decompose P +into segments P1, P2, . . . of length h + 2. Note that a segment Pj = pj0, . . . , pjh+2 yields a +C-gadget of order d and length > h if and only if Tji is either a star or an isolated vertex for +each i ∈ {1, . . . , h + 1}. +Since no such C-gadgets exist by assumption, we obtain that each segment Pj of the path +P contains a vertex pij such that Tij is neither a star nor an isolated vertex. +Assume that Tij is rooted at pij. Since Tij is neither an isolated vertex nor a star, there +must be a (proper) descendant vij of pij (in Tij) such that vij is an (aij, bij)-fork for some +aij, bij ∈ {1, . . . , d}. Now note that there are at most d2 pairs of integers in {1, . . . , d}. Since +we have at least one fork for every segment and since there are at least ⌊k/(h + 2)⌋ > Bd2 + 1 +segments, we thus obtain by the pigeon-hole principle that there is a pair a, b ∈ {1, . . . , d} +such that, for at least B segments Pij, the node vij is an (a, b)-fork in Tij and thus also in +T. Since those forks are pairwise non-adjacent, we obtain, as desired, that the (a, b)-fork +number of T is at least B, concluding Case 1. +Case 2. T has bounded diameter. +Let D be the assumed upper bound on the diameter of trees in T . If T has unbounded +star number then we are finished. Assume instead that T has bounded star number. Then +there is a positive integer s such that for all c ∈ {3, . . . , d} and every tree T ∈ T , Sc(T) < s. +We will show that T has unbounded fork number. Consider any positive integer B. We will +show that there are a, b ∈ {1, . . . , d} and T ∈ T with Fa,b(T) ≥ B. +Let k > (D+1)(Bd2 +1)(d2s+1) be a positive integer. Since T is not matching splittable, +there is a tree T ∈ T whose matching-split number is at least k. Note that T is not a +path since every path with matching-split number at least k has length greater than k > D, +contradicting the bound on the diameter. +Now fix any vertex r of T as the root. Given a vertex v of T, we write Tv for the subtree +rooted at v (assuming that r is the overall root). We call v a rooted fork if Tv is a star — +observe that each rooted fork must indeed be a fork. Let f be the number of rooted forks. +Similar to the argument in Case 1, if f > Bd2 + 1, then by the pigeon-hole principle there +are a, b ∈ {1, . . . , d} such that Fa,b(T) ≥ B. +Hence assume for contradiction that f ≤ Bd2 + 1. Let R be the set of all rays of T and +recall that each ray in R is, by definition, a 2-path of the form R = x0, . . . , xd′ for d′ ≤ d, +where deg(x0) > 2 and xd′ is a leaf. We call a ray R long if d′ ≥ 3. Note that the source of +every ray must either be a rooted fork, or it must lie on a path from the root r to one of the +rooted forks. +Let T ′ be the subtree of T induced by all vertices that lie on paths between r and a +rooted fork (including r and all rooted forks). Since there are f rooted forks and the depth + +18 +Parameterised and Fine-grained Subgraph Counting, modulo 2 +of T is bounded by D, |V (T ′)| ≤ (D + 1)f ≤ (D + 1)(Bd2 + 1). +Consider a vertex v of T ′. Assume for contradiction that v is the source of > ds long +rays (in T). Recall that for all c ∈ {3, . . . , d} we have that Sc(T) < s. Recall further that +each long ray has length d′ for some 3 ≤ d′ ≤ d. Thus we obtain a contradiction by the +pigeon-hole principle. +Now let S be the set containing all vertices of T ′ and all vertices of long rays. Noting +that each long ray has length at most d, and that the source of each long ray must be a +vertex of T ′ by construction, we can use the observation that each vertex of T ′ is the source +of at most ds long rays to (generously) bound +|S| ≤ |V (T ′)| + |V (T ′)| · d · ds . +Note further that T[V (T) \ S] consists only of isolated edges and vertices: The only vertices +in V (T) \ S are non-source vertices of rays of length < 3, the sources of which are in T ′. +Thus, S is a splitting set. Finally, recalling that |V (T ′)| ≤ (D + 1)f ≤ (D + 1)(Bd2 + 1), we +have +|S| ≤ |V (T ′)| + |V (T ′)| · d · ds ≤ (D + 1)(Bd2 + 1)(d2s + 1) , +contradicting the fact that the matching-split number of T is strictly larger than (D + +1)(Bd2 + 1)(d2s + 1). This concludes Case 2, and hence the proof. +◀ +In the next three subsections, we will prove hardness of ⊕Sub(T ) for non-matching- +splittable T in each of the three cases given by Lemma 32. +4.1 +Unbounded C-number +For our hardness proof, it will be useful to find a proper sub-gadget of a C-gadget in a tree. +▶ Definition 33 (Strong C-gadgets, junctions, and closedness). Let C = x0, . . . , xL be a +C-gadget of order d and length L in a tree T. We call C a strong C-gadget with k junctions +if there are integers 0 = i0 < i1 < · · · < ik < ik+1 = L such that +(I) +for all j ∈ {0, . . . , k}, ij+1 − ij > 2d, and +(II) +for all j ∈ {1, . . . , k}, xij is the source of a ray Rj of length d that does not contain one of +the neighbours xij−1 and xij+1 of xij. The vertices xi1, . . . , xik are called the junctions. +Finally, a strong C-gadget is called closed if neither xi1 nor xik are forks.5 +Consider the bottom part of Figure 4 for a visualisation. We start with the following lemma +which establishes the existence of a strong C-gadget with many junctions inside a long enough +C-gadget. +▶ Lemma 34. Let T be a tree such that the longest 2-path in T has length d ≥ 1, and let +k be a positive integer. Then there exists L > 0 (only depending on k and d) such that +the following is true: If T contains an C-gadget of order d and length L, then there exists +1 ≤ d′ ≤ d such that T contains a strong C-gadget of order d′ with at least k junctions. +Proof. Let f(x) = x/(k + 1) − 2d − 1 and let L be large enough such that f d(L) > d. Let +Hd = x0, . . . , xL be a C-gadget of order d and length L in T. +Let d′ = d. Note that Hd′ is a C-gadget of order d′ and length at least L = f d−d′(L) in +T. For each graph Hd′ with d′ ≥ 1 we will either +5 The condition of being closed rules out the special case in which x0 or xL are leaves of T. More generally +it rules out the case where there is a ray from x1 including x0 or from xk including xL. + +L. A. Goldberg and M. Roth +19 +(1) construct a strong C-gadget with k junctions with order d′, or +(2) find a subsequence Hd′−1 of Hd′ that is an C-gadget of order d′ − 1 of length at least +f d−(d′−1)(L). +If we ever do (1) we are finished. If from d′ = 1 we do (2) then we find a 2-path of length at +least f d(L) > d, which is a contradiction. +Here is how we proceed from Hd′ = y0, . . . , yℓ. We set i0 = 0. Then iteratively, for each +j ∈ {1, . . . , k} we will either construct Hd′−1 as in (2) or we find ij ∈ {ij−1 + 2d + 1, . . . , ℓ} +such that yij is the source of a length-d′ ray that does not contain yij − 1 or yij + 1. If we +succeed in defining i1, . . . , ik, ik+1 in this way then y0, . . . , yik+1 is a strong C-gadget with k +junctions of order d′ so (1) is satisfied. +Let us now make this argument rigorous; again, assume that Hd′ = y0, . . . , yℓ is a C- +gadget of order d′ and length ℓ ≥ f d−d′(L). Set i0 = 0 and, starting with j = 0, proceed +iteratively as follows: +1. Let Sj be the set of all indices i ∈ {ij−1 + 2d + 1, . . . , ℓ} such that yi is the source of a +length-d′ ray that does not contain yi−1 and yi+1. +2. If Sj = ∅ then set stop = j and terminate. Otherwise, set ij = min Sj and j ← j + 1, and +go back to 1. +We now distinguish two cases: If stop ≥ k + 1, then we found indices i0, . . . , ik+1 such +that ˆHd′ := y0, . . . , yik+1 is a strong hardness gadget of order d′ with k junctions; hence we +achieved (1) and we are done. Otherwise we have stop < k + 1. Let Ij := {ij, . . . , ij+1 − 1} +for all 0 ≤ j < stop, and let Istop = {istop, . . . , ℓ}. By the pigeon-hole principle, at least one +of those intervals, say Ij′, has size at least ℓ/(stop + 1) ≥ ℓ/(k + 1). Now, by construction of +our iterative procedure above, we find that the sub-interval {ij′ + 2d + 1, . . . , ij′+1 − 1} ⊆ Ij′ +contains no index i such that yi is the source of a length-d′ ray that does not contain yi−1 +and yi+1. Thus, the subsequence Hd′−1 := yij′+2d+1, . . . , yij′+1−1 constitutes a C-gadget +of order d′ − 1. Furthermore, Hd′−1 has length at least ℓ/(k + 1) − 2d − 1 = f(ℓ). Since +ℓ ≥ f d−d′(L), and since f is monotonically increasing, we find that f(ℓ) ≥ f d−(d′−1)(L). +Hence we achieved (2) and we can conclude this case as well. +◀ +Now, by removing the first and the last junction, we can also ensure the existence of a +closed strong C-gadget +▶ Corollary 35. Let T be a tree such that the longest 2-path in T has length d ≥ 1, and +let k be a positive integer. Then there exists L > 0 (only depending on k and d) such that +the following is true: If T contains an C-gadget of order d and length L, then there exists +1 ≤ d′ ≤ d such that T contains a closed strong C-gadget of order d′ with at least k junctions. +Proof. Use Lemma 34 with k + 2 rather than k and observe that every strong C-gadget with +k + 2 junctions also yields a closed strong C-gadget with k junctions by removing i1 and +ik+2 from the list of indices. Since xi1 and xik+2 must have degree at least 3 (they are inner +vertices of a C-gadget and they are junctions), we obtain that neither xi2 and xik+1 can be +forks of T. +◀ +4.1.1 +Constructions of Q and ˆG +For the scope of this subsection, to avoid notational clutter, we assume the following are +given: +Positive integers k and d. + +20 +Parameterised and Fine-grained Subgraph Counting, modulo 2 +A tree T that contains a closed strong C-gadget H = x0, . . . , xℓ of order d with k junctions +xi1, . . . , xik. Additionally, for each j ∈ [k], we fix a ray Rj = xij, r1 +j, . . . , rd +j of length d, +the source of which is xij and which does not contain one of the neighbours xij−1 and +xij+1 — note that the Rj must exist as the xij are junctions. +A k-vertex cubic graph ∆ containing a Hamiltonian cycle v1, . . . , vk, v1. +We emphasise that the set of edges of ∆ not contained in the Hamilton cycle must +constitute a perfect matching, that is, a set of k/2 pairwise non-incident edges. This must +be satisfied since ∆ is cubic. +▶ Definition 36. The core of H, denoted by C(H), contains the subsequence xi1, xi1+1, . . . , xik−1, xik +and the vertices of the rays Rj, that is +C(H) := {xi1, xi1+1, . . . , xik−1, xik} ∪ +k� +j=1 +V (Rj) . +▶ Definition 37 (Q(∆, T, H) and τQ). Set ℓj := ij+1 − ij. The graph Q = Q(∆, T, H) is +obtained from ∆ as follows: +1. The edge {vk, v1} is deleted. +2. For each j ∈ {1, . . . , k − 1} the edge {vj, vj+1} is replaced by a path of length ℓj: +Pj = vj, u1 +j, . . . , uℓj−1 +j +, vj+1 , +where the ut +j are fresh vertices. +3. Each edge e = {vi, vj} not contained on the Hamilton cycle, i.e., j /∈ {i − 1, i + 1}, is +replaced by a path Pi,j of length 2d: +Pi,j = vi, w1 +i , . . . , wd−1 +i +, m(e), wd−1 +j +, . . . , w1 +j, vj , +where the wt +i and wt +j are fresh vertices. +Finally τ = τ(∆, T, H) is a fracture of Q defined as follows: For each m(e), the partition +τm(e) contains two singleton blocks, and for all remaining vertices v of Q the partition τv +only contains one block. +Since ∆, T and H are fixed in this subsection, to avoid notational clutter, we just write Q +and τ, rather than Q(∆, T, H) and τ(∆, T, H). +It turns out that Q is isomorphic to a quotient graph of T[C(H)] obtained by identifying +the endpoints of the rays Ri and Rj for every {vi, vj} ∈ E(∆) with j /∈ {i − 1, i + 1}. This +induces a homomorphism from T[C(H)] to Q that will be useful in the construction of ˆG; +hence we explicitly define this mapping below: +▶ Definition 38 (γ). We define a function γ : C(H) → V (Q) as follows. +1. We map the sequence xi1, xi1+1, . . . , xik−1, xik in C(H) to the sequence v1, . . . , vk in Q. +More precisely, for each j ∈ {1, . . . , k − 1} and t ∈ {1, . . . , ℓj − 1}, we set γ(xij) := vj +and γ(xij+t) := ut +j. +2. For each edge e = {vi, vj} of ∆ with j /∈ {i − 1, i + 1}, we map V (Ri) and V (Rj) to the +path Pi,j. More precisely, for each t ∈ {1, . . . , d − 1} we set γ(rt +i) := wt +i and γ(rt +j) = wt +j. +Furthermore, we set γ(rd +i ) := m(e) =: γ(rd +j ). (Note that the images of the sources of the +rays Ri and Rj are already set in 1.) +▶ Observation 39. The function γ is an edge-bijective homomorphism from T[C(H)] to Q. +Let us provide the induced egde-bijection explicitly: + +L. A. Goldberg and M. Roth +21 +▶ Definition 40. (E′, γE) Define E′ := E(T[C(H)]), that is, E′ ⊆ E(T) contains all +edges on the sub-path xi1, . . . , xik of H and all edges of the rays R1, . . . , Rk. We write +γE : E′ → E(Q) for the edge-bijection from E′ to E(Q) induced by the homomorphism γ. +Now let (G, c) be a Q-coloured graph. We state the following fact explicitly, since it will +be crucial in our construction: +▶ Observation 41. Let (G, c) be a Q-coloured graph. The mapping cE ◦ γ−1 +E +is a map from +E(G) to E′. +Our goal is to construct a graph ˆG = ˆG(G, c, T, H) from G, and an edge-colouring ˆγ : +E( ˆG) �→ E(T) whose range is E(T) such that +⊕Emb((Q +♯ +τ, cτ) → (G, c)) = ⊕ColSub(T → ( ˆG, ˆγ)), +that is, the number of colour-preserving embeddings from the fractured graph Q +♯ +τ to (G, c) +is equal, modulo 2, to the number of subgraphs of ˆG that are isomorphic to T and that +contain each edge-colour in E(T) precisely once. +For what follows, let V (R) := ∪k +j=1V (Rj) be the set of all vertices of the rays R1, . . . , Rk. +We are now able to define ˆG = ˆG(G, c, T, H); the construction is illustrated in Figure 4. +The definition uses the function cE introduced in Definition 6 and the functions γ and γE +introduced in Definitions 38 and 40, respectively. It also uses the mapping cE ◦ γ−1 +E +from +E(G) to E′ (see Observation 41). +▶ Definition 42 ( ˆG(G, c, T, H), ˆγ(G, c, T, H)). Let (G, c) be a Q-coloured graph. The pair +( ˆG, ˆγ) = ( ˆG(G, c, T, H), ˆγ(G, c, T, H)) is an edge-coloured graph constructed as follows, where +the co-domain of ˆγ is E(T): +(A) +The graph ˆG contains G as a subgraph. For each e ∈ E(G), define ˆγ(e) = γ−1 +E (cE(e)). +(B) +The vertex set of ˆG is the union of V (G) and V (T) \ C(H). +(C) +Pairs of vertices in V (T) \ C(H) are connected by an edge in ˆG if and only if they are +adjacent in T. For each such edge e, ˆγ(e) = e. +(D) +The remaining edges of ˆG are defined as follows. For each edge e ∈ E(T) that connects a +vertex z ∈ V (T) \ C(H) to a vertex y ∈ C(H) there are corresponding edges in ˆG. These +edges connect z to all vertices g ∈ V (G) such that c(g) = γ(y) For each such edge e′ in ˆG, +ˆγ(e′) = e. +Observe that for each element Tcol ∈ ColSub(T → ( ˆG, ˆγ)) the induced subgraph +Tcol[G] := Tcol[V (Tcol) ∩ V (G)] +of Tcol is an edge-colourful subgraph in G, that is, Tcol[G] contains precisely one edge per +edge-colour of G under the edge colouring ˆγ hence it contains precisely one edge per edge- +colour of G under cE. As shown in Section 3 in the full version [30] of [31], Tcol[G] thus +induces a fracture ρ = ρ(Tcol) of Q: Two edges {v, w} and {v, y} of Q are in the same block +in the partition ρv corresponding to vertex v of Q if and only if the edges of Tcol[G] that are +coloured γ−1 +E ({v, w}) and γ−1 +E ({v, y}) are adjacent. In what follows, we show that ρ must +always be equal to τ(∆, T, H) (see Definition 37). +▶ Lemma 43. For every Tcol ∈ ColSub(T → ( ˆG, ˆγ)) we have that ρ(Tcol) = τ(∆, T, H). +Proof. To avoid notational clutter, we set ρ := ρ(Tcol) and τ := τ(∆, T, H). Let T1 and T2 +be the subtrees of T attached to the ends of the C-gadget H as shown in the bottom part of +Figure 4. + +22 +Parameterised and Fine-grained Subgraph Counting, modulo 2 +Figure 4 (Below): The tree T containing a closed strong C-gadget of order d; the green dashed +lines are rays of length d. (Left): The construction of ˆG = ˆG(G, c, T, H); note that the removal of +the vertices and edges coloured blue yields G (see Definition 42), and note that G is Q-coloured as +depicted. (Right): The graphs ∆ and Q = Q(∆, T, H); we assume in the picture that {v2, vk−1} is +an edge of ∆. +We first give an overall intuition of the proof; consider Figure 5 for an illustration. Since +Tcol is isomorphic to T, there must be a (unique) path connecting T1 and T2 in ˆG (recall + +L. A. Goldberg and M. Roth +23 +that, since Tcol is edge-colourful and since every edge in T1 and T2 has a different colour — +see (C) in Definition 42 — Tcol must contain all edges in T1 and T2). We claim that this +path must follow the outer cycle in ˆG, in which case the designated rays in R of length d +at the junctions must follow the inwards direction and thus induce τ. To see why the path +connecting T1 and T2 must follow the outer cycle, first recall that Vj is the subset of V (G) +coloured by c with vj. Then recall that the path between Vj and Vj+1 along the outer cycle +in ˆG has length ℓj ≥ 2d + 1. Hence the designated rays in R cannot be used to cover all +edge colours in the path between Vj and Vj+1. +We next provide a rigorous argument. Let +S := V (T1) ∪ V (T2) ∪ {x0, . . . , xi1−1} ∪ {xik+1, . . . , xk+1}. +Note that S is a subset of V (T) \ V (H) hence it is a subset of V (T) and of V ( ˆG). +We first claim that every fork and every ray of length > d of T must be fully contained +in the subgraph of T induced by S. This claim follows from the definition of closed strong +C-gadgets. In particular, the condition of being closed implies that neither xi1 nor xik is a +fork. +As a consequence, every fork and every ray of length greater than d of Tcol must be +contained in the subgraph of ˆG induced by S as well. Additionally, this implies that none of +the vertices in Tcol[G] can be a fork or the source of a ray of length > d in Tcol — otherwise, +Tcol would have either more forks or more rays of length > d than T, contradicting the fact +that Tcol and T are isomorphic. +Recall that V1, . . . , Vk denote the subsets of vertices of G that are coloured by c with +v1, . . . , vk. Now let P be the (unique) path P in Tcol that connects T1 with T2. Then, starting +with V1 and ending with Vk, the path P must pass through a sequence of colour classes +V1 = Vj1, Vj2, . . . , Vjt = Vk of G. The following claim formalises the idea that this sequence +must correspond to the Hamilton cycle v1, . . . , vk in ∆. +Claim: +We have t = k and Vji = Vi for each i ∈ [k]. +Before proving the claim, we show that it implies the lemma. Since, from the claim, P +must follow the outer cycle, the fracture ρ = ρ(Tcol) induced by Tcol must split the inner +paths of length 2d (otherwise Tcol would contain a cycle). However, since there are no sources +or rays of length greater than d outside of S in Tcol, ρ must split all of the inner length-2d +paths at the central vertex m(e). Furthermore, it cannot split additional vertices since this +would disconnect Tcol. Thus, ρ is the fracture τ, concluding the proof. ■ +To conclude the proof, we now prove the claim. Note first that P cannot pass through +any of the colour classes Vi more than once as this would cause Tcol to use an edge-colour +multiple times. Next assume for contradiction that P misses some colour class Va for some +a ∈ [2, k − 1] (i.e., we assume that t < k). Since Tcol is a connected tree containing all of the +edge colours in Q there must be an index ji ̸= a and a vertex u ∈ Vji ∩ P such that Tcol +contains a (unique) path Pu from u to a vertex w ∈ Va. In order to get the contradiction, +root Tcol at u. Construct a subtree Tcol(u) of Tcol as follows: For each neighbour x of u +except the ancestor of w on the path from u, we delete x and all of its descendants. Observe +that the edge colours of Tcol(u) are disjoint from the edge-colours of P and that V (Tcol(u)) +is disjoint from S. Now, if Tcol(u) is a path, then (using that ℓi > 2d), we obtain that u is +the source of a ray in Tcol of length greater than d, contradicting the fact that every ray of +length > d of Tcol is in the subgraph of ˆG induced by S. Otherwise, Tcol(u) contains a fork, +contradicting the fact that all forks of Tcol are in the subgraph of ˆG induced by S. +Having established that t = k and that no Vi is visited more than once, it remains to +show that P visits the colour classes in the correct order, that is Vji = Vi for each i ∈ [k]. + +24 +Parameterised and Fine-grained Subgraph Counting, modulo 2 +Figure 5 Illustration of Lemma 43: The only possibility for an edge-colourful copy of T to be +embedded in ˆG is depicted in red. +Assume for contradiction that this is not the case, which allows us to set +m := min{i ∈ [k] | Vji ̸= Vi} − 1 . +Note that m ≥ 1 since j1 = 1. Let zm ∈ Vm ∩ P and zm+1 ∈ Vm+1 ∩ P and recall that G +contains colour classes U 1 +m, . . . , U ℓm−1 +m +corresponding to the path +Pm = vm, u1 +m, . . . , uℓm−1 +m +, vm+1 + +L. A. Goldberg and M. Roth +25 +in Q (see Definition 37). Let us now define the subtrees Tcol(m) and Tcol(m + 1): +For Tcol(m) we root Tcol at zm and for each neighbour x of zm in Tcol, we delete x and all +of its descendants unless x ∈ U 1 +m. +For Tcol(m + 1) we root Tcol at zm+1 and for each neighbour x of zm+1 in Tcol, we delete +x and all of its descendants unless x ∈ U ℓm−1 +m +. +Note that at least one of Tcol(m) and Tcol(m + 1) must have depth greater than d (if rooted +at zm and zm+1, respectively), since ℓm > 2d and Tcol is edge-colourful with respect to ˆγ, +that is, we have to make sure that we cover all of the edge colours +{vm, u1 +m}, {u1 +m, u2 +m}, . . . , {uℓm−1 +m +, vm+1} +Finally, regardless of which one of the two subtrees has depth greater than d, we will find +either a fork, or the source of a ray of length greater than d outside of the set S, yielding +the desired contradiction and concluding the proof of the claim, and hence the proof of the +lemma. +◀ +We are now able to prove the main lemma of this subsection. +▶ Lemma 44. ⊕Emb((Q +♯ +τ, cτ) → (G, c)) = ⊕ColSub(T → ( ˆG, ˆγ)). +Proof. We start with the following claim from [30]. +Claim: A colour-preserving embedding ϕ ∈ Emb((Q +♯ +τ, cτ) → (G, c)) is uniquely defined +by its image (which is a subgraph of (G, c)). +For convenience, we provide a proof of the claim: Consider in image (G′, c′) of ϕ where G′ +is a subgraph of G and c′ = c |V (G′). Let e = {u, v} be an edge of G′ Then c′(e) = {c(u), c(v)} +is an edge of Q since c is a Q-colouring. Recall that Q +♯ +τ is Q-coloured by the function +cτ that maps wB to w for each w ∈ V (Q) and block B ∈ τw. Now recall the definition +of fractured graphs (Definition 8) and let B1 and B2 be the blocks of τc(u) and τc(v) that +contain c(e). Then, since ϕ is an embedding, it maps c(u)B1 to u and c(v)B2 to v. Since Q +does not have isolated vertices, continuing this process over all edges of G′ defines ϕ. This +concludes the proof of the claim. ■ +By the claim, it is sufficient to construct a bijection b from elements in ColSub(T → ( ˆG, ˆγ)) +to subgraphs (G′, c′) that are images of embeddings in Emb((Q +♯ +τ, cτ) → (G, c)). Given +Tcol ∈ ColSub(T → ( ˆG, ˆγ)) we set b(Tcol) := (Tcol[G], c(Tcol)) where c(Tcol) is the colouring of +vertices of Tcol[G] which agrees with ˆγ on the edges of Tcol[G]. In the rest of the proof, we +show that b is the desired bijection. +First, we have to show that for all Tcol, (Tcol[G], c(Tcol)) is the image of an embedding in +Emb((Q +♯ +τ, cτ) → (G, c)). To this end, recall that Tcol[G] induces a fracture ρ = ρ(Tcol) of Q. +By the definition of ρ, Tcol[G] and Q +♯ +ρ are isomorphic and this isomorphism preserves the +colours so cρ agrees with ˆγ on the edges of Q +♯ +ρ. This implies that cρ and c(Tcol) are the +same. So (Tcol[G], c(Tcol)) is the image of an embedding in Emb((Q +♯ +ρ, cρ) → (G, c)). Finally, +Lemma 43 guarantees that ρ = τ. +Second, we will show that b is injective. To this end, let Tcol1 ̸= Tcol2 ∈ ColSub(T → ( ˆG, ˆγ)). +Since Tcol1 and Tcol2 must both fully contain V (T) \ C(H), and since both are edge-colourful +(see Definition 42), the only possibility for Tcol1 and Tcol2 not being equal is that they disagree +on G, that is, Tcol1[G] ̸= Tcol2[G]. This proves b to be injective. +Finally, we will show that b is surjective: Given any (G′, c′) that is the image of an +embedding ϕ ∈ Emb((Q +♯ +τ, cτ) → (G, c)), we construct Tcol(G′, c′) ∈ ColSub(T → ( ˆG, ˆγ)) +with b(Tcol(G′, c′)) = (G′, c′) as follows. Observe first that G′ is isomorphic to T[C(H)] since + +26 +Parameterised and Fine-grained Subgraph Counting, modulo 2 +Q +♯ +τ is, by definition of τ, isomorphic to T[C(H)]: Splitting the inner paths of length 2d in +Q at their central vertices yields precisely T[C(H)]. Then Tcol(G′, c′) is obtained by adding +the remainder of T to (G′, c′): +1. We add to (G′, c′) all vertices in V (T) \ C(H) (see (B) in Definition 42). +2. We add all edges between vertices in V (T) \ C(H) that are present in ˆG (see (C) in +Definition 42). +3. Finally, we connect a vertex in z in V (T) \ C(H) with a vertex w in G′ if and only if z +and w are connected in ˆG (see (D) in Definition 42). +The resulting subgraph Tcol(G′, c′) of ˆG is clearly edge-colourful and isomorphic to T, +concluding the proof. +◀ +We are now able to establish hardness of ⊕Sub(T ) in case of unbounded C-number. +▶ Lemma 45. Let T be a recursively enumerable class of trees of unbounded C-number. +Then ⊕Sub(T ) is ⊕W[1]-hard. +Proof. Assume first that T contains trees with 2-paths of unbounded length. In this case +we reduce from the problem of counting k-cycles, modulo 2, which was shown ⊕W[1]-hard +in [11]. In the first step, this problem reduces to the problem of counting s-t-paths of length +k, modulo 2 as shown in Lemma 5.2 in the full version [28] of [27]. In the second and final +step, we can easily reduce from the problem of counting s-t-paths of length k, modulo 2, to +⊕Sub(T ), as shown in Figure 6: Concretely, let (G, s, t, k) be a problem instance. Since T +contains trees with 2-paths of unbounded length, we can find, in time only depending on k, +a tree T in T containing a 2-path x0, x1, . . . , xk+1, xk+2 of length k + 2. Let furthermore T1 +and T2 be the subtrees of T as depicted in Figure 6. We construct a graph G′ from G in two +steps as follows: First, we add fresh vertices x0 and xk+2 and edges {x0, s} and {t, xk+2}. +Second, we add T1 and T2 and identify their roots with x0 and xk+2, respectively. The +construction is depicted in Figure 6 as well. Now let A be the set of subgraphs of G′ that are +isomorphic to T and that contain all edges of T1 and T2. It is easy to see that the cardinality +of A is equal to the number of s-t-paths of length k in G. Thus it suffices to compute |A| +mod 2, using an oracle for ⊕Sub(T ). This can be achieved by a simple application of the +inclusion-exclusion principle: Setting S = E(T1) ∪ E(T2), we have +|A| = +� +J⊆S +(−1)|J| · #Sub(T → G′ \ J) , +(7) +where G′ \ J is the graph obtained from G′ by deleting all edges in J. We can conclude the +reduction by observing that the number of terms in (7) only depends on T and thus on k, +and that our oracle to ⊕Sub(T ) allows us to evaluate (7) modulo 2. +For the remainder of the proof we can thus assume that the length of any 2-path in any +tree in T is bounded by a constant d. Since T has unbounded C-number, we obtain that the +trees in T contain C-gadgets of order d of unbounded length. By Corollary 35 we obtain +that for any positive integer k, there is a value d′ in the range 1 ≤ d′ ≤ d such that there is +a tree Tk in T which contains a strong C-gadget of order d′ with k junctions. +Let C be a class of cubic Hamiltonian graphs of unbounded treewidth. Assume w.l.g. +that, for each k, the class C contains at most one graph with k vertices; otherwise we just +keep one k-vertex graph with the largest treewidth among all k-vertex graphs in C. For each +∆ ∈ C set T∆ := T|V (∆)|, that is T∆ is contained in T and contains a strong C-gadget H∆ +with at least |V (∆)| junctions. Recall Definition 37 and set +Q := {Q(∆, T∆, H∆) | ∆ ∈ C} . + +L. A. Goldberg and M. Roth +27 +Observe that Q(∆, T∆, H∆) contains as minor the graph obtained from ∆ by removing one +edge. Since the removal of a single edge can decrease the treewidth only by a constant, and +since treewidth is minor-monotone, we have that Q has unbounded treewidth. +By Theorem 12 the problem ⊕cp-Hom(Q) is therefore ⊕W[1]-hard. Thus it suffices to +show that +⊕cp-Hom(Q) ≤fpt +T ⊕Sub(T ) . +In the first step, we reduce the computation of ⊕Hom((Q, idQ) → ⋆) to the computation +of ⊕Emb((Q +♯ +τ, cτ) → ⋆); here, τ is the fracture defined in Definition 37. To this end, it was +shown in [29] that +⊕Emb((Q +♯ +τ, cτ) → ⋆) = +� +ρ≥τ +µ(τ, ρ) · ⊕Hom((Q +♯ +ρ, cρ) → ⋆) , +(8) +where the relation “≥” and the Möbius function µ are over the lattice of fractures. We +omit introducing these objects in detail, since we only require that the coefficient of the +term ⊕Hom((Q +♯ +⊤, c⊤) → ⋆) (which is equal to ⊕Hom((Q, idQ) → ⋆)) in the above linear +combination was shown in [29] to be equal to +� +v∈V (Q) +(−1)|τv|−1 · (|τv| − 1)! . +Since each partition τv has at most two blocks, the above term is odd. Thus, by Lemma 14, we +can evaluate the term ⊕Hom((Q +♯ +⊤, c⊤) → ⋆) if we can evaluate the entire linear combination, +that is, if we can evaluate ⊕Emb((Q +♯ +τ, cτ) → ⋆). It thus remains to show how we can evaluate +⊕Emb((Q +♯ +τ, cτ) → ⋆) using our oracle for ⊕Sub(T ). +To this end, we use Lemma 44: Given any Q = Q(∆, T∆, H∆)-coloured graph (G, c) +for which we want to compute ⊕Emb((Q +♯ +τ, cτ) → (G, c)), we first construct ( ˆG, ˆγ) as in +Definition 42. Then Lemma 44 yields that +⊕Emb((Q +♯ +τ, cτ) → (G, c)) = ⊕ColSub(T∆ → ( ˆG, ˆγ)). +Finally, by Lemma 15 we can compute ⊕ColSub(T∆ → ( ˆG, ˆγ)) in FPT time using an +oracle for ⊕Sub(T∆ → ⋆). Since the size of T∆ only depends on Q, and since, with input Q +we can find T∆ (recall that T is recursively enumerable) this yields indeed a parameterised +Turing-reduction and the proof is concluded. +◀ +4.2 +Unbounded Star Number +We will use the same strategy as in Subsection 4.1: Given a tree T with large star number, +we start with a properly chosen cubic graph ∆, and we construct a graph Q depending on ∆ +and T which contains ∆ as a minor. Then we show that for any Q-coloured graph (G, c), +we can construct an edge-coloured graph ( ˆG, ˆγ) such that ⊕ColSub(T → ( ˆG, ˆγ)) is equal to +⊕Emb((Q +♯ +τ, cτ) → (G, c)) for a particular fracture τ. +To this end, let T be a tree with star number (at least) 6k for some positive integer k. By +definition of the star number, there is a d ≥ 3 such that T contains a vertex s which is the +source of 6k rays R1, . . . , R6k of length precisely d. For each i ∈ [6k], let Ri = s, r1 +i , . . . , rd +i . +Furthermore, let Ts be the subtree of T obtained by deleting the vertices r1 +i , . . . , rd +i for each +i ∈ [6k]; consider Figure 7 for an illustration. + +28 +Parameterised and Fine-grained Subgraph Counting, modulo 2 +Figure 6 Reduction from counting s-t-paths of length k, modulo 2, in a graph G to counting +copies of a tree T with a 2-path of length at least k + 2. +Figure 7 A tree with Sd(T) ≥ 6k. +▶ Definition 46 (Q). Let ∆ be cubic graph on k vertices. We obtain Q from ∆ by substituting +each vertex v by a gadget depicted in Figure 8. Afterwards, we connect the gadgets as follows: +If {v, x} is an edge of ∆, then we identify the vertex vx in the gadget of v and the vertex xv +in the gadget of x. +▶ Observation 47. ∆ is a minor of Q. +The fracture τ of Q that we will be interested in is defined as follows; Figure 9 depicts +the fractured graph Q +♯ +τ. +▶ Definition 48 (τ). Let Q be the graph defined in Definition 46. +For each edge {v, x} of ∆, the graph Q contains a vertex vx(= xv), which has degree 2. +We let τvx be the partition consisting of 2 singleton blocks. +For each vertex v of ∆, the vertices v3 and v5 have degree 2 in Q. We let τv3 and τv5 be +the partitions consisting of 2 singleton blocks. +For each vertex v of ∆, the vertices v2, v4 and v6 have degree 3 in Q. For each i ∈ {2, 4, 5} +we let τvi be the partition consisting of one block of size 2 corresponding to the edges +incident to vi from the left and the right, and one block of size 1 corresponding to the +edge incident to vi from below. + +L. A. Goldberg and M. Roth +29 +Figure 8 The construction of Q; the vertices v1, . . . , v6 on the gadget of v are emphasized. +Figure 9 Illustration of the fractured graph Q +♯ +τ via fracturing the vertex gadgets. +For all other vertices u of Q, we let τu be the partition consisting only of one block. +Analogously to the notion of a core in the case of unbounded C-number, we will identify +a specific subgraph of the tree T and we will use it to define the graph ˆG later. +▶ Definition 49 (V ′). Let V ′ be the vertex subset of T defined as follows: +V ′ := +� +� � +i∈[6k] +V (Ri) +� +� \ {s} . +Furthermore, we set E′ := E(T[V ′]). +Observe that T[V ′] is a (disjoint) union of 6k paths of length d − 1, where the vertices of +the i-th path are r1 +i , . . . , rd +i . Observe further that V (T) = V (Ts) ˙∪V ′ and that +E(T) = E′ ˙∪ E(Ts) ˙∪ {{s, r1 +i } | i ∈ [6k]} . +(9) +Next, note that the edges of Q can be decomposed into 6k paths, each of length d − 1: +There are k vertices of ∆. For each vertex v ∈ V (∆) the graph Q contains, by definition, a +gadget corresponding to v, the edges of which can be decomposed into 6 paths P 1 +v , . . . , P 6 +v +of length d − 1 (formally, the fractured graph Q +♯ +τ yields precisely this decomposition; see +Figure 9). Additionally, for each v ∈ V (∆) and i ∈ [6], the first vertex of P i +v is chosen to be +vi as depicted in Figure 8. + +30 +Parameterised and Fine-grained Subgraph Counting, modulo 2 +▶ Definition 50 (γ, γE). We define a function γ : T[V ′] → V (Q) as follows. Recall that T[V ′] +is the union 6k paths P ′ +j := r1 +j, . . . , rd +j for j ∈ [6k]. Fix any bijection b : [6k] → V (∆) × [6]. +Then γ maps P ′ +j to P i +v, where b(j) = (v, i). In particular, we enforce that the first vertices +of the paths are mapped onto each other, that is, γ(r1 +j) := vi. Additionally, we define +γE : E′ → E(Q) by mapping e to γ(e). +▶ Observation 51. The function γ is an edge-bijective homomorphism from T[V ′] to Q. +Specifically, γE is a bijection. +Now let (G, c) be a Q-coloured graph. We state the following explicitly, since it will be +crucial in our reduction. +▶ Observation 52. Let (G, c) be a Q-coloured graph. The mapping cE ◦ γ−1 +E +is a map from +E(G) to E′. +Let us now construct a graph ˆG from a Q-coloured graph G; an illustration is provided +in Figure 10. +▶ Definition 53 (( ˆG, ˆγ)). Let (G, c) be a Q-coloured graph. The graph ˆG is an edge-coloured +graph, with colouring ˆγ : E( ˆG) → E(T), constructed as follows: +(A) +The graph ˆG contains G as a subgraph. For each e ∈ E(G) we set ˆγ(e) = γ−1 +E (cE(e)). +(B) +The vertex set of ˆG is the union of V (G) and V (Ts), and pairs of vertices in V (Ts) are +connected by an edge in ˆG if and only they are adjacent in T. For each such edge e, +ˆγ(e) = e. +(C) +The remaining edges of ˆG are defined as follows. For each edge e = {s, r1 +j} ∈ E(T), we +connect s to all vertices in G that are coloured (by c) with γ(r1 +j) (see Definition 50), and +for each of those newly added edges e′ we set ˆγ(e′) := e +Observe that ˆγ colours the edges of ˆG with E(T); the cases (A), (B), and (C) correspond, +respectively, to the sets E′, E(Ts) and {{s, r1 +i } | i ∈ [6k]} (see Equation (9)). Similarly to +the case of unbounded C-gadgets, for each element Tcol ∈ ColSub(T → ( ˆG, ˆγ)) the induced +subgraph +Tcol[G] := Tcol[V (Tcol) ∩ V (G)] +of Tcol is an edge-colourful subgraph in G, that is, Tcol[G] contains precisely one edge per +edge-colour of G under the edge colouring ˆγ hence it contains precisely one edge per edge- +colour of G under cE. As shown in Section 3 in the full version [30] of [31], Tcol[G] thus +induces a fracture ρ = ρ(Tcol) of Q: Two edges {v, w} and {v, y} of Q are in the same block +in the partition ρv corresponding to vertex v of Q if and only if the edges of Tcol[G] that are +coloured γ−1 +E ({v, w}) and γ−1 +E ({v, y}) are adjacent. In what follows, we show that ρ must +always be equal to τ(∆, T, H) (see Definition 48). +▶ Lemma 54. For every Tcol ∈ ColSub(T → ( ˆG, ˆγ)) we have that ρ(Tcol) = τ. +Proof. Let Tcol ∈ ColSub(T → ˆG, ˆγ). Since Tcol must include each of the edge colours given +by ˆγ (precisely) once, we have that Tcol must fully contain Ts. Note that Ts fully contains T +except for 6k rays of length d, and the only way to attach those rays in ˆG is via the vertex s. +Now consider the subgraph Tcol[G + s] of Tcol defined as follows: +Tcol[G + s] := Tcol[(V (Tcol) ∩ V (G)) ∪ {s}] . +Since Tcol includes all edge colours given by ˆγ, we have that s must have degree 6k in +Tcol[G + s]: By (C) in Definition 53, the vertex s must be connected (within Tcol[G + s]) to +one vertex in each of the colour classes Vi = c−1(vi) for v ∈ V (∆) and i ∈ [6]. Additionally, +this implies the following: + +L. A. Goldberg and M. Roth +31 +Figure 10 The construction of ˆG. The graph G within ˆG is depicted in black. + +32 +Parameterised and Fine-grained Subgraph Counting, modulo 2 +▶ Observation 55. Tcol[G + s] is isomorphic to the d-stretch of K1,6k with s at the centre. +In the remainder of the proof, we will show that the only way for Tcol to (colourfully) +embed the 6k rays of length d is as depicted in Figure 11. Note that this will conclude the +proof since the induced fracture of the depicted embedding is τ. +Hence we proceed with proving the claim. We first consider, for each edge {v, x} ∈ E(∆), +the vertex vx = (xv) of Q (see Definition 46 and Figure 8). The vertex vx has two neighbours +nv and nx in Q, where nv denotes the neighbour in the gadget of v and nx denotes the +neighbour in the gadget of x. Recall that we write Vx = c−1(vx), Nv = c−1(nv), Nx = +c−1(nx) ⊆ V (G) for their colour class within G (and thus within ˆG). Since Tcol is edge- +colourful, it must contain precisely one edge ev between Vx and Nv and one edge ex between +Vx and Nx (see (A) in Definition 53). Now observe that every vertex in Vx has distance (at +least) d to s within ˆG. This has two crucial consequences: +First, the endpoints of ev and ex inside Vx cannot be equal: Otherwise, they could not be +part of a ray of length precisely d with source s, and this would contradict the previous +observation that Tcol[G + s] is isomorphic to the d-stretch of K1,6k with s at the centre +(Observation 55). +Hence, second, the endpoints of ev and ex inside Vx both have degree 1. Consequently, +they must be the endpoints of two of the rays of length d. However, the only way for this +to be true is them each being connected to s as depicted in Figure 11; in all other cases, +Tcol[G + s] cannot be isomorphic to the d-stretch of K1,6k with s at the centre. +The second consequence implies that the edge colours corresponding to the edges in the paths +P 2 +v , P 4 +v , and P 6 +v are covered for each v (recall that Tcol must include each edge colour precisely +once). Thus, the only possibility to include the remaining edge colours corresponding to the +paths P 1 +v , P 3 +v , and P 5 +v while keeping Tcol[G + s] being isomorphic to the d-stretch of K1,6k, is +to embed, for each gadget, the remaining 3 rays of length d as depicted in Figure 11. This +concludes the proof. +◀ +We are now able to prove the main lemma of this section. +▶ Lemma 56. ⊕Emb((Q +♯ +τ, cτ) → (G, c)) = ⊕ColSub(T → ( ˆG, ˆγ)). +Proof. Thanks to Lemma 54, the proof is similar to the proof of Lemma 44: Colour- +preserving embeddings in Emb((Q +♯ +τ, cτ) → (G, c)) are uniquely identified by their image, +and a bijection b from ColSub(T → ( ˆG, ˆγ)) to images of colour-preserving embeddings in +Emb((Q +♯ +τ, cτ) → (G, c)) is given by b : Tcol �→ Tcol[G]. +◀ +Similarly to the proof in Section 4.1, Lemma 56 is sufficient for hardness. +▶ Lemma 57. Let T be a recursively class of trees of unbounded star number. Then ⊕Sub(T ) +is ⊕W[1]-hard. +Proof. The proof is almost identical to the proof of Lemma 45, with the exception that we +use Q, τ, ˆG, and ˆγ as defined in the current section, and that we rely on Lemma 56 for the +identity +⊕Emb((Q +♯ +τ, cτ) → (G, c)) = ⊕ColSub(T → ( ˆG, ˆγ)). +The remainder of the proof transfers verbatim. +◀ + +L. A. Goldberg and M. Roth +33 +Figure 11 Illustration of the unique way to colourfully embed T into ˆG. The induced fracture +is τ. + +34 +Parameterised and Fine-grained Subgraph Counting, modulo 2 +4.3 +Unbounded Fork number +We will rely on the same high-level strategy as the one that we used when the C-number +or star number was unbounded: Given a tree T with large a-b-fork number, we start +with a properly chosen cubic graph ∆, and we construct a graph Q which depends on T +and ∆, and which contains ∆ as a minor. Afterwards, we show that for any Q-coloured +graph (G, c) we can construct an edge-coloured graph ( ˆG, ˆγ) where the co-domain of ˆγ is +E(T) such that #ColSub(T → ( ˆG, ˆγ)) is equal (modulo 2) to #Emb((Q +♯ +τ, cτ) → (G, c)) for +a particular fracture τ of Q. However, proving this equality will be more involved than +it was in the previous cases: In Sections 4.1 and 4.2, we were able to prove, implicitly, +that #ColSub(T → ( ˆG, ˆγ)) = #Emb((Q +♯ +τ, cτ) → (G, c)), that is, we were able to establish +equality, rather than equality modulo 2. In the current case, we are not able to prove equality +and must therefore rely on parity arguments, which makes the case slightly more involved. +We start by fixing the following: +Positive integers k, a and b with a ≤ b and k ≥ 2. +A tree T with Fa,b(T) ≥ 2k. By definition of forks (Definition 29), T contains designated +sources s1 +1, s2 +1, . . . , s1 +k, s2 +k such that for each (i, j) ∈ [k] × [2], the source sj +i is the source of +two (distinct) rays Fa(i, j) of length a and Fb(i, j) of length b. Additionally degNL(sj +i) = 1. +We assume w.l.o.g. that the designated sources are ordered by their leaf-degrees, that is +degL(s1 +1) ≥ degL(s2 +1) ≥ · · · ≥ degL(s1 +k) ≥ degL(s2 +k) . +(10) +Consider Figure 12 for an illustration of T, its designated sources, and the rays Fa(i, j) +and Fb(i, j). +A k-vertex bipartite cubic graph ∆ with vertices V (∆) = {v1, . . . , vk}. +A proper 3-edge-colouring C : E(∆) → {s, m, ℓ} of ∆.6 +We first note that, since there are at least 2k ≥ 4 sources in T, any pair of distinct sources +must not be adjacent: Otherwise, the tree T would either be disconnected, or one of the +sources would have degNL at least 2, both of which is a contradiction. +▶ Observation 58. For any distinct pair (i, j) ̸= (i′, j′) we have that sj +i and sj′ +i′ are not +adjacent in T. +Next, we define the graph Q. +▶ Definition 59 (Q). The graph Q is obtained from ∆ and C via substituting vi by the +gadget depicted in Figure 13 for each i ∈ [k]. Afterwards, for every edge e = {vi, vj} of ∆ we +identify the vertex coloured with C(e) in the gadget of vi with the vertex coloured with C(e) +in the gadget of vj. +While Definition 59 will be useful in our proofs, we note the following easier equivalent +way to define Q. +▶ Observation 60. The graph Q is obtained from ∆ and C by substituting each edge of +colour s (of ∆) with a path of length 2a, each edge of colour m with a path of length 2b, and +each edge of colour ℓ with a path of length 2(a + b). Consequently, ∆ is a minor of Q. +The fracture τ of Q that we will be interested in is defined as follows; Figure 14 depicts +the fractured graph Q +♯ +τ. +6 That is, C(e1) ̸= C(e2) whenever e1 ̸= e2 share a vertex. Note that every cubic bipartite graph has a +3-edge-colouring by Hall’s Theorem. + +L. A. Goldberg and M. Roth +35 +Figure 12 A tree T with Fa,b(T) ≥ 2k. Note that the parents of the sj +i are not necessarily +distinct. The rays Fa(i, j) and Fb(i, j) are depicted in red. +v1 +i +v2 +i +a +ℓ +b +s +a +m +b +Figure 13 A vertex gadget in the construction of Q in Definition 59. A dashed line labelled with +a (resp. b) depicts a path of length a (resp. b). + +36 +Parameterised and Fine-grained Subgraph Counting, modulo 2 +Figure 14 The fractured graph Q +♯ +τ. Note that the illustration only depicts the fracturing of a +single vertex gadget. +▶ Definition 61 (τ). Let Q be the graph defined in Definition 59. +For each edge e = {vi, vj} of ∆, there is a vertex C(e) ∈ {s, m, ℓ} of degree 2 that connects +the gadgets of vi and vj. We let τC(e) be the partition consisting of two singleton blocks. +For each vertex vi of ∆, the gadget of vi in Q contains the vertex v1 +i of degree 3 which is +connected to s via a path of length a, to m via a path of length b, and to ℓ via a path of +length a + b. Let es, em, and eℓ be the first edges on those paths. We set +τvi = {{es, em}, {eℓ}} . +For all other vertices u of Q, we let τu be the partition consisting only of one block. +Next we identify specific substructures of T that will be necessary in the construction of ˆG. +▶ Definition 62. Recall that sj +i with (i, j) ∈ [k] × [2] are the designated sources of T. +T ′ is the graph obtained from T by deleting, for each (i, j) ∈ [k] × [2], the designated +source sj +i as well as all rays with source sj +i. +For each (i, j) ∈ [k] × [2], pj +i is the neighbour of sj +i which is not contained in a ray. Note +that pj +i is unique by definition of forks. Note that pj +i ∈ V (T ′) and that the pj +i are not +necessarily pairwise distinct. +For each (i, j) ∈ [k] × [2], dj +i = degL(sj +i) − 2, that is, dj +i is the number of rays with source +sj +i minus 2. Note that dj +i ≥ 0 since each sj +i is the source of Fa(i, j) and Fb(i, j). +F := +� +(i,j)∈[k]×[2] +(Fa(i, j) ∪ Fb(i, j)) , that is, F is the subset of V (T) that contains the +vertices of the rays Fa(i, j) and Fb(i, j) (which includes sj +i) for each (i, j) ∈ [k] × [2]. +E′ := E(T[F]). +An illustration of these notions is given in Figure 12. + +L. A. Goldberg and M. Roth +37 +Observe that T[F] is a disjoint union of 2k paths of length a + b. Specifically, for each +(i, j) ∈ [k] × [2] it contains the path +F j +i := T[Fa(i, j) ∪ Fb(i, j)] . +It turns out that Q is isomorphic to a quotient graph of T[F], since for each vertex vi of ∆, +the vertex gadget of vi decomposes into two paths of length a + b. In fact, this decomposition +is given by the fractured graph Q +♯ +τ (see Figure 14). Formally, we have the following: +▶ Observation 63. T[F] ∼= Q +♯ +τ ∼= 2kPa+b. +Similarly to the previous two cases, we introduce functions γ and γE which we will need +for defining the edge-colours of ˆG. +▶ Definition 64 (γ, γE). We define a function γ : F → V (Q) as follows: +1. For each i ∈ [k], γ maps F 1 +i to the (a + b)-path in the gadget of vi from s to m, such that +γ(s1 +i ) = v1 +i . +2. For each i ∈ [k], γ maps F 2 +i to the (a + b)-path in the gadget of vi from v1 +i to ℓ, such that +γ(s2 +i ) = v2 +i . +Furthermore, we write γE : E′ → E(Q) by setting γE({x, y}) := {γ(x), γ(y)}. +Note that the definition of γE is well-defined since γ is a homomorphism by Observation 63. +Concretely, γ can be viewed as the composition of an isomorphism from T[F] to Q +♯ +τ and +the Q-colouring cτ of Q +♯ +τ (see Definition 9). Furthermore, γE is clearly a bijection. Hence, +similarly to the previous sections, we point out the following: +▶ Observation 65. Let (G, c) be a Q-coloured graph. The mapping cE ◦ γ−1 +E +is a map from +E(G) to E′. +We are now able construct a graph ˆG from a Q-coloured graph G; an illustration is +provided in Figure 15. +▶ Definition 66 (( ˆG, ˆγ)). Let (G, c) be a Q-coloured graph. The pair ( ˆG, ˆγ) is an edge-coloured +graph constructed as follows, where the co-domain of ˆγ is E(T). +(A) +The graph ˆG contains G as a subgraph. For each e ∈ E(G), define ˆγ(e) = γ−1 +E (cE(e)). +(B) +The vertex set of ˆG is the union of V (G) and V (T) \ F. +(C) +Pairs of vertices in V (T)\F are connected by an edge in ˆG if and only if they are adjacent +in T. For each such edge e, we set ˆγ(e) = e. +(D) +The remaining edges of ˆG are defined as follows. For each edge e ∈ E(T) that connects a +vertex z ∈ V (T) \ F to a vertex y ∈ F there are corresponding edges in ˆG. These edges +connect z to all vertices g ∈ V (G) such that c(g) = γ(y) For each such edge e′ in ˆG, +ˆγ(e′) = e. +In (D), the only edges in T connecting z ∈ V (T) \ F to a vertex y ∈ F satisfy that y is one +of the designated sources sj +i, and z is either pj +i ∈ V (T ′) or z is contained in one of the dj +i +rays with source sj +i that are not Fa(i, j) or Fb(i, j) (see Definition 62). +Similarly to the other cases, for each element Tcol ∈ ColSub(T → ( ˆG, ˆγ)) the induced +subgraph Tcol[G] := Tcol[V (Tcol) ∩ V (G)] of Tcol is an edge-colourful subgraph in G. Also, +Tcol[G] induces a fracture ρ = ρ(Tcol) of Q as follows. First, recall that G is Q-coloured by c, +and that G is contained in ˆG (see (A) in Definition 66). Next note that Tcol[G] is a subgraph +of G that contains each edge colour in the image of cE ◦ γ−1 +E +precisely once. Since γE is a +bijection from E′ to E(Q), we can thus equivalently view Tcol[G] as a subgraph of G that +contains each edge colour in the image of cE precisely once. This fact allows us to define +ρ(T ) in terms of the function cE as follows. + +38 +Parameterised and Fine-grained Subgraph Counting, modulo 2 +Figure 15 The graph ˆG. Depicted in the centre is the part of G (within ˆG) that is coloured with +the vertices of the i-th vertex gadget of Q. Depicted in black are the subtree T ′ of T (left), and, +as dashed lines, the inner edges of the d1 +i + d2 +i rays incident to s1 +i and s2 +i (right) — here, the inner +edges are those that are not incident to the sources s1 +i and s2 +i . Each edge of ˆG fully contained in the +black part has a unique colour w.r.t. ˆγ (see Definition 66 (C)). Pairs consisting of remaining edges +have the same colour (w.r.t. ˆγ) if and only if they are depicted with the same colour. + +L. A. Goldberg and M. Roth +39 +Figure 16 Illustration of the condition that yields invalid trees at (i, 1) (below) and (i, 2) (above). +Edges contained in E′ are coloured red. +▶ Definition 67 (ρ(Tcol)). Let Tcol be an element of ColSub(T → ( ˆG, ˆγ)). +The fracture +ρ = ρ(Tcol) of Q is defined as follows. Two edges {v, w} and {v, y} of Q are in the same +block in the partition ρv corresponding to vertex v of Q if and only if the edges of Tcol[G] that +are coloured by cE with {v, w} and {v, y} are incident. +With ( ˆG, ˆγ) defined, we can finally state formally the goal of this section. Recall that +(G, c) is a Q-coloured graph. +▶ Lemma 68. Suppose that |c−1(v)| is odd for each v ∈ V (Q). Then ⊕ColSub(T → ( ˆG, ˆγ)) = +⊕Emb((Q +♯ +τ, cτ) → (G, c)). +The proof requires some additional set-up. In particular, we need the condition that +|c−1(v)| is odd to deal with the case in which what we call “invalid trees” arise. To this +end, recall that V j +i = c−1(vj +i ) denotes the set of vertices in G that are coloured by c with vj +i . +Since G is a subgraph of ˆG (see Definition 66), we slightly abuse notation and write V j +i also +for the subset of vertices in ˆG corresponding to V j +i in G. +▶ Definition 69. Let Tcol ∈ ColSub(T → ( ˆG, ˆγ)) and let (i, j) ∈ [k] × [2]. We call Tcol invalid +at (i, j) if the following two conditions are met: +(I) +Tcol contains precisely two vertices x and y in V j +i . +(II) +x is adjacent to pj +i and not incident in Tcol to any edge coloured with a colour in E′ (see +Definition 66 (A)). +Otherwise Tcol is called valid at (i, j). We call Tcol an invalid tree if there exists a pair +(i, j) ∈ [k] × [2] such that Tcol is invalid at (i, j). Otherwise, we call Tcol a valid tree. We +write ColSubval(T → ( �G, ˆγ)) for the set of all valid Tcol in ColSub(T → ( �G, �γ)). +Consider Figure 16 for an illustration of Definition 69. +▶ Lemma 70. Suppose that |c−1(v)| is odd for each v ∈ V (Q). Then the number of invalid +trees Tcol ∈ ColSub(T → ( ˆG, ˆγ)) is even. + +40 +Parameterised and Fine-grained Subgraph Counting, modulo 2 +Proof. For the proof, given two tuples (i, j) and (i′, j′) in [k] × [2] we write (i′, j′) < (i, j) +if (i′, j′) is lexicographically smaller than (i, j). +Write T (i, j) for the set of all Tcol ∈ +ColSub(T → ( ˆG, ˆγ)) that are invalid at (i, j) but valid on all pairs (i′, j′) < (i, j). We will +prove that T (i, j) is even for all (i, j) ∈ [k] × [2]; this is sufficient for the lemma to hold. +Hence fix (i, j), let Tcol ∈ T (i, j), and let x and y be as in Definition 69. Since V j +i = c−1(vj +i ) +and for j ∈ [2], vj +i is a vertex of Q, the assumption in the statement of the lemma implies that +|V j +i | is odd. Since x and y are distinct vertices in V j +i , V j +i contains additional vertices other +than x and y. Fix a vertex x′ ∈ V j +i \ {x, y}. Obtain T ′ +col from Tcol by deleting x (including +edges incident to x) and by adding x′ and the edge {x′, u} for every u that was adjacent to +x — this is well-defined since x is not incident to any edge coloured with a colour in E′, and +by construction of ˆG (see Definition 66 (C) and (D)) whenever {x, u} ∈ E( ˆG) is an edge not +coloured with a colour in E′, then {x′, u} ∈ E( ˆG) for every x′ ∈ V j +i . Additionally, {x, u} +and {x′, u} have the same edge-colour. Hence, clearly, T ′ +col an edge-colourful subgraph of ˆG +that is isomorphic to Tcol (and thus to T). For this reason, we obtain that T ′ +col ∈ T (i, j). +More generally, the observation that T ′ +col ∈ T (i, j) allows us to define an equivalence +relation on T (i, j): Let Tcol and T ′ +col be elements of T (i, j), and let x and x′ be the vertices +in Tcol and T ′ +col that satisfy (II) in Definition 69. We set Tcol and T ′ +col to be equivalent if and +only if one can obtained from the other by switching x with x′ as defined above. The size of +one equivalence class is precisely |V j +i | − 1 = |c−1(vj +i )| − 1, which is even by the premise of +the lemma. +◀ +For the proof of Lemma 68, we need to establish some facts about rays and 2-paths of +elements Tcol ∈ ColSubval(T → ( ˆG, ˆγ)), which are those Tcol ∈ ColSub(T → ( ˆG, ˆγ)) that are +valid. We encapsulate these facts in the next section. +4.3.1 +The Proof of Lemma 68 +We first note that, thanks to Lemma 70, it suffices to prove that +#ColSubval(T → ( ˆG, ˆγ)) = #Emb((Q +♯ +τ, cτ) → (G, c)) . +This requires some preparation. We first fix the following objects (recall the definitions of +2-path, Definition 27 and ray, Definition 28). +Tcol is an element of ColSubval(T → ( ˆG, ˆγ)) +Tcol[G] is the graph obtained from Tcol[V (Tcol)∩V (G)] with isolated vertices removed. (In +fact, our proof will show that, for valid trees Tcol ∈ ColSubval(T → ( ˆG, ˆγ)), the induced +subgraph Tcol[V (Tcol) ∩ V (G)] cannot have isolated vertices. However, at the current +point of the proof, it is easiest to just remove them.) +For any positive integer t, Rt is the set of length-t rays in T. Pt is the set of length-t +2-paths in T that are not rays. +For any positive integer t, Rt +col is the set of length-t rays in Tcol and Pt +col is the set of +2-paths in Tcol that are not rays. Note that |Rt| = |Rt +col| and |Pt| = |Pt +col| for all t since +T and Tcol are isomorphic. +We will also rely on the following notion of external rays and 2-paths. +▶ Definition 71. A 2-path P of Tcol is called external if the following two conditions are +satisfied. +Except for the endpoints, none of the vertices of P is contained in V (G). +P does not contain an edge of G. + +L. A. Goldberg and M. Roth +41 +Definition 71 applies whether or not P is a ray. The following lemmas establish that all +2-paths of Tcol of length greater than b must be external. +▶ Lemma 72. Suppose that t is an integer that is greater than b. Suppose that, for all t′ > t, +every 2-path in Rt′ +col ∪ Pt′ +col is external. Then every 2-path in Rt +col ∪ Pt +col is external. +Proof. We first construct a bijection f from Rt to Rt +col. +We will use this bijection to +argue that every ray in Rt +col is external. In order to define the bijection, consider a ray +R = r0, r1, . . . , rt in Rt. Since t > b ≥ a, R is not one of the designated rays Fa(i, j) and +Fb(i, j). If r0 is not among the designated sources sj +i, then, by the construction of ˆG, R +is contained in T ′ and thus R ∈ Rt +col. In this case R is external and we set f(R) := R. +Alternatively, suppose that r0 = sj +i for some i and j. Then R must be one of the dj +i black +rays in Figure 12 (see Definition 62). By the construction of ˆG and the fact that Tcol is +edge-colourful, there is a vertex x ∈ V j +i such that Tcol contains the path x, r1, . . . , rt. In +Tcol, as in T, the vertices r1, . . . , rt−1 have degree 2 and the vertex rt has degree 1. Vertex +x cannot have degree 1 in Tcol since this would disconnect Tcol. Also, vertex x x cannot +have degree 2: To see this, assume for contradiction that x has degree 2. Then there is an +integer t′ > t and a ray R′ ∈ Rt′ +col the last vertices of which are x, r1, . . . , rt. Since x is not an +endpoint of the ray and since x ∈ V (G), the ray R′ is not external, contradicting the premise +of the lemma. Hence x has degree at least 3 and therefore f(R) := x, r1, . . . , rt is an external +ray of Tcol. The function f is injective by construction. Since Tcol and T are isomorphic, +|Rt| = |Rt +col| and thus f is a bijection. Since the image of f only contains external rays, we +have shown that every element of Rt +col is external. +Every ray in the image of f has the property that its degree-1 endpoint is not contained +in V (G). Since the image of f is Rt +col, we obtain +(∗) Every ray in Rt +col has the property that its degree-1 endpoint is not contained in V (G). +To complete the proof, we show that every 2-path in Pt +col is external. Following the same +strategy that we used before, we construct a bijection g from Pt to Pt +col. Every 2-path in +the range of g is external, so we will conclude that every element of Pt is external. In order +to define the bijection, consider a 2-path P = p0, . . . , pt in Pt. If neither of the endpoints +of P is among the designated sources sj +i, then P is contained in T ′ and thus P ∈ Pt. In +this case, P is external and we set g(P) := P. If exactly one endpoint of P is among the +designated sources, say p0 = sj +i, then there is a vertex x ∈ V j +i such that x, p1, . . . , pt is a +path in Tcol. The vertices p1, . . . , pt−1 have degree 2 in Tcol (as in T) and the vertex pt has +degree at least 3. +If x has degree 1 in Tcol, the ray R = pt, . . . , p1, x is in Tcol, and its degree-1 endpoint x +is in V (G), contradicting (∗). Hence x cannot have degree 1 in Tcol. Similarly, x cannot +have degree 2, since this would create a 2-path longer than t in Tcol that is not external, +which contradicts the premise of the lemma. +Hence x has degree at least 3, and thus +g(P) := x, p1, . . . , pt is an external 2-path in Pt +col. +For the last case, suppose that both endpoints of P are among the designated sources, +say p0 = sj +i and pt = sj′ +i′ . Then there are x and y in, respectively, V j +i and V j′ +i′ such that +x, p1, . . . , pt−1, y is a path in Tcol. Again, p1, . . . , pt−1 must all have degree 2 in Tcol as well. +We show that both x and y have degree at least 3 in Tcol: If both have degree 1, then +Tcol is disconnected. If one of them has degree 1 and the other one has degree at least 3, +then we created a ray of length t whose degree-1 endpoint in in V (G), contradicting (∗). +If one has degree 1 and the other one has degree 2, then we found a ray longer than t +which is not external, contradicting the premise of the lemma. If one has degree 2 and +the other has degree at least 2, then there is a non-external 2-path longer than t, again + +42 +Parameterised and Fine-grained Subgraph Counting, modulo 2 +contradicting the premise of the lemma. Thus, as desired, both must have degree at least 3. +Therefore, g(P) := x, p1, . . . , pt−1, y is an external 2-path in Pt +col. The function g is injective +by construction. Since Tcol and T are isomorphic, |Pt| = |Pt +col| and thus g is a bijection. +Since the image of g only contains external 2-paths, we have shown that every element of +Pt +col is external, concluding the proof. +◀ +▶ Lemma 73. Suppose that t is an integer that is greater than b. Then every 2-path in +Rt +col ∪ Pt +col is external. +Proof. Let tmax be the maximum integer for which Rtmax ∪ Ptmax is nonempty. Let Φt be the +proposition “t ≤ b or every 2-path in Rt +col ∪ Pt +col is external”. +We will show by induction on tmax−t that Φt holds. The base case arises when tmax−t = 0, +so t = tmax. If tmax ≤ b then Φt is satisfied. Otherwise, for each t′ > t, the set Rt′ +col ∪ Pt′ +col is +empty and we can invoke Lemma 72 to conclude that Φt holds. +For the induction step, consider t such that tmax − t ≥ 1. By the induction hypothesis, +Φt′ holds for all t′ ∈ {t + 1, . . . , tmax}. If t ≤ b then Φt holds. Otherwise, for all t′ > t > b, +we know from Φt′ that every 2-path in Rt′ +col ∪ Pt′ +col is external. We can then apply Lemma 72 +to conclude that every 2-path in Rt +col ∪ Pt +col is external. +◀ +Before proceeding with the proof of Lemma 68, we provide an overview of the central +steps of the proof. Recall that it suffices to prove that +#ColSubval(T → ( ˆG, ˆγ)) = #Emb((Q +♯ +τ, cτ) → (G, c)) +and that we have a fixed an element Tcol of ColSubval(T → ( ˆG, ˆγ)) and proved various +properties about it. +(1) Our goal is to show that Tcol is embedded in ˆG in the following manner (see Figure 17). +For each (i, j) ∈ [k] × [2], Tcol contains a ray Ra(i, j) of length a and a ray Rb(i, j) of +length b; those rays correspond to the designated rays Fa(i, j) and Fb(i, j) in T (recall +that T and Tcol are isomorphic.) +a. T ′ is part of Tcol. +b. For every i ∈ [k] and j ∈ [2], the vertices pj +i in T ′ is connected to a vertex wj +i of G +with c(wj +i ) = vj +i = γ(sj +i). In Tcol, the vertex wj +i is the source of dj +i rays other than +Ra(i, j) and Rb(i, j). The vertices of these dj +i rays are not in T ′ and are not in G. +The edge colours of the edges in these rays in ˆγ are the same as the edge-names in T +(see Definition 66 (C)). +c. The length-a ray Ra(i, 1) is a path in Tcol from w1 +i to the vertex ua(i, 1) of G with +some colour c(ua(i, 1)) (a vertex of Q). This colour c(ua(i, 1)) corresponds to the +vertex “s” in the gadget of the vertex vi of ∆ (see Definition 59 and Figure 13). +d. The length-b ray Rb(i, 1) is a path in Tcol from w1 +i to the vertex ub(i, 1) of G with +some colour c(ub(i, 1)) (a vertex of Q). This colour c(ub(i, 1)) corresponds to the +vertex “m” in the gadget of the vertex vi of ∆ (see Definition 59 and Figure 13). +e. The length-b ray Rb(i, 2) is a path in Tcol from w2 +i to the vertex ub(i, 2) of G with +some colour c(ub(i, 2)) (a vertex of Q). This colour c(ub(i, 2)) corresponds to the +vertex “ℓ” in the gadget of the vertex vi of ∆ (see Definition 59 and Figure 13). +f. The length-a ray Ra(i, 2) is a path in Tcol from w2 +i to the vertex ua(i, 2) ̸= w1 +i of G +with some colour c(ua(i, 2)) = γ(s1 +i ) = v1 +i (recall that the colour is a vertex of Q). +g. For every edge e = {vi, vi′} in ∆, ua(i, 1) ̸= ua(i′, 1), ub(i, 1) ̸= ub(i′, 1) and ub(i, 2) ̸= +ub(i′, 2). + +L. A. Goldberg and M. Roth +43 +(2) We now make some observations about the fracture ρ = ρ(Tcol) from Definition 67, given +that Tcol is embedded in ˆG as described in Item (1). +The definition of Q (Definition 59) tells us that, for every edge e = {vi, vi′} in ∆, +there is a degree-2 vertex y of Q that connects the gadgets of vi and vi′. Vertex y +corresponds to the vertex C(e) ∈ {s, m, ℓ} in the two gadgets. Suppose without loss of +generality that C(e) = s. The other cases are similar. From (1c) the colour C(e) = s +is the same as c(ua(i, 1)) and c(ua(i′, 1)). From (1b) c(w1 +i ) = v1 +i and c(w1 +i′) = v1 +i′. +Since Tcol is colourful and the embedding is as in (1), the edges of the ray from w1 +i to +ua(i, 1) have different edge colours to the ray from w1 +i′ to ua(i′, 1). Thus, the edge in +G in the first ray that is adjacent to ua(i, 1) (call it ei) has a different colour from the +edge n G in the second ray that is adjacent to ua(i′, 1) (call it ei′). Concretely, we +have cE(ei) = {s, x} and cE(ei′) = {s, x′} where x and x′ are the neighbours of s (in +Q) in the gadgets of vi and vi′, respectively. By (1g) we have ua(i, 1) ̸= ua(i′, 1) and +thus, by definition of ρ (Definition 67), ρy consists of two singleton blocks. Similar +arguments show that ρ coincides with τ (see Definition 61) at every vertex of Q that +corresponds to vertex “s”, “ℓ” or “m” in any gadget corresponding to any vertex vi +of ∆. +We now continue with the vertices v1 +i for i ∈ [k] of Q. See Figure 13 for the gadget +containing v1 +i in Q and Figure 17 for the graph ˆG. We will use “s”, “ℓ” and “m” as +the names of these vertices in the gadget containing v1 +i . The vertex v1 +i has degree +3 and is connected to s via a path of length a, to m via a path of length b and to +ℓ via a path of length a + b. Let ys, ym, and yℓ be the successors of v1 +i on those +paths, that is, the edges incident to v1 +i in Q are es := {v1 +i , ys}, em := {v1 +i , ym}, an +eℓ := {v1 +i , yℓ}. Now, by (1c) and (1d), the edges of Tcol that are coloured (by cE) with +es and em are {w1 +i , ra} and {w1 +i , rb}, where ra and rb are the successors of w1 +i on the +rays Ra(i, 1) and Rb(i, 1), respectively. Furthermore, by (1f), the edge of Tcol that +is coloured (by cE) with eℓ is {ua(i, 2), ˆr} where ˆr is the vertex in the ray Ra(i, 2) +that is adjacent to ua(i, 2). Since ua(i, 2) ̸= w1 +i (by (1f)), the edge {ua(i, 2), ˆr} is not +incident to either {w1 +i , ra} or {w1 +i , rb}. Thus ρv1 +i = {{es, em}, {eℓ}} which coincides +with τv1 +i by Definition 61. So τ and ρ coincide at vertex v1 +i . +Next are the vertices v2 +i for i ∈ [k] (see Figure 13). This case is easy. If Tcol is +embedded as described in (1) (see Figure 17), then, for each i ∈ [k], there is only one +vertex of Tcol which is coloured by c with colour v2 +i . This vertex is w2 +i . Thus every +edge of Tcol whose edge colour includes v2 +i is incident to w2 +i . Hence ρv2 +i only consists +of one block, which coincides with τv2 +i by Definition 61. +Finally, every remaining vertex of Q (see Figure 13) has degree 2. Let y be such a +vertex and let y1 and y2 be the neighbours of y. Then the edges of Tcol coloured by cE +with {y, y1} and {y, y2} must be successive edges on one of the rays Ra(i, 1), Rb(i, 1), +Ra(i, 2), or Rb(i, 2). So these successive edges are both incident to the vertex of the +ray that is coloured y by c. Thus ρy only consists of one block, which coincides with +τy. +Since we have shown that the fractures ρ and τ coincide at every vertex of Q, we conclude +that ρ = τ. +(3) We next explain why it is useful to have ρ = τ. +Recall that our goal is to prove +that #ColSubval(T → ( ˆG, ˆγ)) = #Emb((Q +♯ +τ, cτ) → (G, c)) and that Tcol is an element +of ColSubval(T → ( ˆG, ˆγ)). Our method will be to show that the function β defined by + +44 +Parameterised and Fine-grained Subgraph Counting, modulo 2 +β(Tcol) = Tcol[G] is a bijection from ColSubval(T → ( ˆG, ˆγ)) to Emb((Q +♯ +τ, cτ) → (G, c)). +It will turn out that this implies that the embedding ρ coincides with τ. +(4) In order to prove Item (1) we will proceed as follows. +(i) We show that all 2-paths (including rays) of Tcol are external, except for 2k rays of +length b and 2k rays of length a. Note that we already established this claim for +2-paths of lengths greater than b in Lemma 73. +(ii) Then we show that Tcol contains two degree-1 vertices in each of the vertex sets L +and M of G (within ˆG) — see Figure 17, recalling that, for each vertex gadget, the +sets L and M denote the vertex subsets of G that are coloured by c with ℓ and m. +The point of this is that we will also prove that Tcol has two degree-1 vertices in S +(Item 4iv) — this will split off the part of Tcol corresponding to a single gadget, so +we will only have to study the embedding of Tcol within each gadget. We prove +the claim about L and M by using the fact that Tcol is isomorphic to T and that +all 2-paths longer than b are external. This implies that if vi and vi′ are the two +vertices of ∆ sharing this gadget then the 2-paths between V 2 +i and V 2 +i′ are covered +by two rays in Tcol, both of which end in L. +(iii) We next show that the degree-1 vertices in (4ii) are the endpoints of 2k rays of +length b. We have already seen that for each of the k gadgets the endpoints of +these rays are in L and M. For the i’th gadget, the sources are in V 1 +i and V 2 +i If +b > a then we show that all remaining 2-paths of length b and also all 2-paths with +lengths in a + 1, . . . , b − 1 are external. The proof of this claim relies on the same +arguments as the proof of Lemma 73. +(iv) Next, we show that for each gadget, Tcol contains two degree-1 vertices in S — see +Figure 17. The proof uses the fact that all 2-paths longer than a that are not +covered by (4iii) are external. +(v) We next show that the degree-1 vertices in (4iv) are the endpoints of 2k rays of +length a. We have already seen that for each of the k gadgets the endpoints of +these rays are in S. For the i’th gadget, the source is in V 1 +i . +(vi) The remaining details of the proof rely on the fact that the tree Tcol is valid. +We now provide the proof in detail; for convenience, we also restate the lemma. +▶ Lemma 68. Suppose that |c−1(v)| is odd for each v ∈ V (Q). Then ⊕ColSub(T → ( ˆG, ˆγ)) = +⊕Emb((Q +♯ +τ, cτ) → (G, c)). +Proof. We will prove that for any Tcol ∈ ColSubval(T → ( ˆG, ˆγ)), Item (1) of the proof overview +holds. +Using this fact and the argument from Item (2) of the proof overview, we conclude that +for any Tcol ∈ ColSubval(T → ( ˆG, ˆγ)), ρ(Tcol) = τ. +Recall that every edge-colourful subgraph of G induces a fracture of Q. +Let G′ be an element of Emb((Q +♯ +τ, cτ) → (G, c)). +This means that G′ is an edge- +colourful subgraph of G that induces τ. We wish to see how G′ can be extended to some +Tcol +′ ∈ ColSubval(T → ( ˆG, ˆγ)). We know from Item (1) that any Tcol +′′ ∈ ColSubval(T → ( ˆG, ˆγ)) +can only be embedded in ˆG in one way, so G′ can only be extended in one way. The details +are as follows. We claim that there is only one possible extension because T ′ has to be +included and item (b) of (1) ensures that, for each j ∈ [2], the vertex pj +i is connected to wj +i . +The rest of (1) shows the unique way to include the rays, so the extension is unique. +Let β be the function from ColSubval(T → ( ˆG, ˆγ)) that maps any element Tcol to Tcol[G]. +Note that Tcol[G] ∈ Emb((Q +♯ +τ, cτ) → (G, c)) since ρ(Tcol) = τ and ρ(Tcol) is a function of + +L. A. Goldberg and M. Roth +45 +Figure 17 An embedding Tcol of T in ˆG that yields the fracture τ. We will show that this is the +only way to embed T in ˆG in such a way that each edge-colour is used precisely once. Note that +dashed lines depict paths in Tcol, and solid lines depict edges in Tcol. +Tcol[G]. Let β′ be the function that maps an element of Emb((Q +♯ +τ, cτ) → (G, c)) to its unique +extension in ColSubval(T → ( ˆG, ˆγ)). Note that β◦β′ and β′◦β are both the identity. Therefore +β is a bijection and |ColSubval(T → ( ˆG, ˆγ))| = |Emb((Q +♯ +τ, cτ) → (G, c))|.The lemma follows +from Lemma 70. +To finish the proof, we will fix Tcol ∈ ColSubval(T → ( ˆG, ˆγ)) and we will show that Item (1) +of the proof overview holds. Part (a) of (1) is trivial since Tcol is edge-colourful so it contains +T ′. The first sentence of (b) is also trivial. We will next focus on (c)–(g), noting along the +way when the rest of (b) is proved. +Recall from Definition 59 that, for each i ∈ [k], the graph Q contains +for each vertex vj such that ∆ has an edge e = {vi, vj} with C(e) = m, a path Pi,j of +length 2b from v1 +i to v1 +j , and +for each vertex vj such that ∆ has an edge e = {vi, vj} with C(e) = ℓ, a path Pi,j of +length 2b from v2 +i to v2 +j . + +46 +Parameterised and Fine-grained Subgraph Counting, modulo 2 +Recall from Definition 6 that cE maps edges of G to edges of Q. Furthermore, G is a +subgraph of ˆG, see Definition 66 (A). Let Tcol(i, j) be the subgraph of Tcol[G] induced by +edges e of G such that cE(e) is in the path Pi,j +By construction, Tcol(i, j) is the union of some number of paths. We will next argue that +it is the union of exactly two disjoint length-b paths: +If Tcol(i, j) has more than two components then at least one component is disconnected +from T ′ in Tcol, contradicting the fact that Tcol is a tree. +If Tcol(i, j) is a single path then it is contained in a 2-path of length at least 2b. Since +this 2-path contains an edge in G, it is not external (Definition 71). This contradicts +Lemma 73. +If Tcol(i, j) is the union of exactly two disjoint paths, one of which has length larger +than b then this larger 2-path is contained in a 2-path that is not external contradicting +Lemma 73 +What we have shown is that T(i, j) consists of two length-b paths. For some t ∈ {1, 2}, +one of these paths is from V t +i and the other is from V t +j . To be more precise and to fix the +notation for t = 1, we have now shown that, for each i ∈ [k], Tcol[G] contains a path Rb(i, 1) +of length b that starts at a vertex w1 +i ∈ V 1 +i . We refer to the other end of this path as ub(i, 1). +The vertex ub(i, 1) has degree 1 and is contained in M (i.e., in c−1(m)). We next argue that +w1 +i has degree at least 3 in Tcol. (See Figure 18.) +If w1 +i has degree 1 in Tcol then Tcol is disconnected, contradicting the fact that it is a tree. +If w1 +i has degree 2 in Tcol, then Tcol has a ray of length at least b + 1 that is not external, +which is again a contradiction. +By the same reasoning, Tcol contains a ray Rb(i, 2) of length b that starts at a vertex w2 +i ∈ V 2 +i +and ends at a vertex ub(i, 2). The ray Rb(i, 2) is contained in Tcol[G]. +We have just finished parts (d) and (e) of (1) and the part of (g) that concerns length b. +So what we have shown corresponds to Figure 18. We would now like to prove parts (c) and +(f) but unfortunately these are more difficult because we have to show where the rays with +lengths between a and b are embedded so that we can argue about where the length-a rays +are embedded. +Define ˆR := �k +i=1{Rb(i, 1), Rb(i, 2)}. Recall that k, a, and b are positive integers with +a ≤ b and k ≥ 2 and that T has Fa,b(T) ≥ 2k and Tcol ∼= T. Also, Rb +col is the set of length-b +rays in Tcol and Rb is the set of length-b rays in T. (See Figure 12.) Using the notation that +we have established, we will prove the following claims. +Claim 1: Let P ∈ (Rb +col \ ˆR) ∪ Pb +col. If a < b then P is external. +We prove Claim 1 for the case where P ∈ Rb +col \ ˆR. The other case is similar but easier. +Observe that |Rb| ≥ 2k since Fa,b(T) ≥ 2k. So Rb can be partitioned as follows +Rb[S] is the set of the 2k length-b rays Fb(i, j) whose sources are s1 +1, . . . , s2 +k and which +are depicted as red dashed lines in Figure 12. +Rb[T] = Rb \ Rb[S] contains the remaining rays of length b. +Our goal is to show that all rays in Rb +col \ ˆR are external. To do this, we first show that +|Rb[T]| = |Rb +col \ ˆR| and we then provide an injection from Rb[T] to Rb +col \ ˆR in which all +elements of the range are external rays. +To show that |Rb[T]| = |Rb +col \ ˆR|, first note that |Rb| = |Rb +col| because T and Tcol are +isomorphic. We further have |Rb[S]| = | ˆR| = 2k. +We next define the (injective) map from Rb[T] to Rb +col\ ˆR . For any ray R = r0, r1, . . . , rb ∈ +Rb[T] we proceed as follows. + +L. A. Goldberg and M. Roth +47 +Figure 18 Illustration of the embedding of Tcol after the rays of length b are analysed. Solid +lines depict edges, dashed lines depict paths, and dash-dotted lines depict sequences of edges (the +identification of the endpoints of which we have not yet been determined). Note that both Rb(i, 1) +and Rb(i, 2) must be of length b. Except for those two rays, the identification of endpoints of +the remaining edges that are incident to G (within ˆG) has not been determined yet either; this is +depicted by the dotted circles inside the colour classes. The fracture ρ induced by Tcol will depend +on the identification of the edges of Tcol, both endpoints of which lie in G. The goal is to show that +the endpoints have to be identified precisely as depicted in Figure 17. + +48 +Parameterised and Fine-grained Subgraph Counting, modulo 2 +If r0 is not among the designated sources sj +i, then R is fully contained in T ′ (see Figure 12) +and thus R is a ray in Tcol. We map R to itself. Note that R is external since it is fully +contained in T ′. +Otherwise, r0 = sj +i and R is one of the rays depicted as black dashed lines in Figure 12. +Since Tcol is edge-colourful, and by construction of ˆG, Tcol contains a path R′ = x, r1, . . . , rb +where x ∈ V j +i . (See Figure 15.) If x has degree 1 in Tcol then Tcol is disconnected, which +is not true. If x has degree 2 in Tcol then Tcol has a non-external ray which is longer +than b, which is also a contradiction by Lemma 73. Thus, x has degree at least 3 in Tcol, +and R′ is an external ray. We map R to R′. +This concludes the proof of Claim 1 for the case where P ∈ Rb +col \ ˆR. ■ +Claim 2: Suppose that there is an integer t′ such that a < t′ < b. Suppose that P ∈ Rt′ +col∪Pt′ +col. +Then P is external. +In order to explain the proof of Claim 2, recall that we have established the following +facts about 2-paths in Tcol in Lemma 73 and Claim 1. +Every 2-path of length greater than b is external. +Every 2-path of length b is either a ray in ˆR or is external. +With those 2-paths covered, the proof of Claim 2 is analogous to the proof of Lemma 73. ■ +Using Claims 1 and 2 we will now prove parts (c) and (f) of (1). For each 2-path whose +length is larger than a, we have already shown that it is in ˆR or we have shown that it is +external. In order to prove (c) we will show that, for each edge {vi, vi′} of ∆ with colour s, +the sequence of edges in Tcol between V 1 +i and V 1 +i′ is the union of two disjoint length-a rays. +This is formalised as follows. +Note that for each edge {vi, vj} of ∆ coloured by the 3-edge-colouring C with s, there is +a path Pi,j of length 2a from v1 +i to v1 +j . Recall that cE maps edges of G to edges of Q. We +write Tcol(i, j) for the subgraph of Tcol[G] induced by edges e of G such that cE(e) is in the +path Pi,j. By construction, Tcol(i, j) is the union of some number of paths. We will next +argue that it is the union of exactly two disjoint length-a paths: +If Tcol(i, j) has more than two components then at least one component is disconnected +from T ′ in Tcol, contradicting the fact that Tcol is a tree. +If Tcol(i, j) is a single path then it is contained in a 2-path of length at least 2a. Since +this 2-path contains an edge in G, it is not external (Definition 71). Additionally, it is +not included in ˆR. This contradicts the aforementioned fact that each 2-paths of length +at least a + 1 is external or included in the set ˆR. +If Tcol(i, j) is the union of exactly two disjoint paths, one of which has length larger +than a, then this larger path yields a contradiction similarly to the previous case. +What we have shown is that T(i, j) consists of two length-a paths. One of these paths +is from V 1 +i and the other is from V 1 +j . To be more precise and to fix the notation, we have +now shown that, for each i ∈ [k], Tcol[G] contains a path Ra(i, 1) of length a that starts at a +vertex ˆw1 +i ∈ V 1 +i . We refer to the other end of this path as ua(i, 1). The vertex ua(i, 1) has +degree 1 and is contained in S (i.e., in c−1(s)). So we have established Part (c) of item (1). +Consider Figure 19 for an illustration of all the information we gathered so far. (The vertices +labelled zj +i and the edge set Ea +i in the figure will be discussed below). +To finish the proof of item (1) we will show part (f) and the rest of part (b). We take these +together. Recall that for every i ∈ [k] there is a path P a +i = v1 +i , y1, . . . , ya−1, v2 +i of length a in +Q from v1 +i to v2 +i . Since Tcol is edge-colourful, it includes each of the colours of the edges on +this path exactly once — these colours are γ−1 +E ({v1 +i , y1}),γ−1 +E ({y1, y2}), . . . ,γ−1 +E ({ya−1, v2 +i }). + +L. A. Goldberg and M. Roth +49 +Figure 19 Depiction of the embedding of Tcol as established after Claim 2 (in the proof of +Lemma 68). +Solid lines depict edges, dashed lines depict paths, and dash-dotted lines depict +sequences of edges (the identification of the endpoints of which has not yet been determined). Note +that we have not yet determined how the endpoints inside of the colour classes V 1 +i and V 2 +i are +identified either; this is depicted by the dotted circles inside these colour classes. Proving that +the embedding of Tcol is as depicted in Figure 17 requires us to show that all endpoints in V 2 +i are +identified, and that all endpoints in V 1 +i , except for x1 +i , are identified. + +50 +Parameterised and Fine-grained Subgraph Counting, modulo 2 +Under the edge colouring cE, the same edges of Tcol are coloured with the colours {v1 +i , y1}, +{y1, y2}, . . . , {ya−1, v2 +i }. +Let e1, . . . , ea be the edges of Tcol with those colours; we write Ea +i for this set of edges +(as is depicted in Figure 19). We let x1 +i be the vertex of Tcol which is contained in V 1 +i and +incident to e1, and we let x2 +i be the vertex of Tcol which is contained in V 2 +i and incident to ea. +Let z1 +i and z2 +i be the vertices of Tcol in V 1 +i and V 2 +i that are adjacent to p1 +i and p2 +i — those +vertices are depicted in Figure 19 and we point out that, a priori, x1 +i might be equal to to z1 +i +and x2 +i might be equal to z2 +i . +Claim 3: There are no vertices in V (Tcol) ∩ V 1 +i other than z1 +i , x1 +i , w1 +i , ˆw1 +i and vertices in +the d1 +i rays. +To prove Claim 3, assume for contradiction that z is such a vertex. Recall that V 1 +i is an +independent set (because vertices in V 1 +i all receive the same colour under c.) Since Tcol is +connected, z has a neighbour outside of V 1 +i but all of the edge colours incident to V 1 +i are +already used. ■ +The proof of the following claim is similar. +Claim 4: There are no vertices in V (Tcol) ∩ V 2 +i other than z2 +i , x2 +i , w2 +i , and vertices in the d2 +i +rays. ■ +Claim 5: Both z1 +i and z2 +i have degree at least 3 in Tcol. We prove the claim for z1 +i ; an +analogous argument applies for z2 +i . Assume first for contradiction that z1 +i has degree 1. Since +Tcol is connected, Claim 2.5 implies that |V (Tcol)∩V 1 +i | = 2 so x1 +i = w1 +i = ˆw1 +i and the depicted +vertices in the d1 +i rays are also identified with this vertex. By Definition 69, Tcol is invalid, +giving a contradiction. +Now assume for contradiction that z1 +i has degree 2. We consider two subcases: +z1 +i is identical to x1 +i . Then Tcol is disconnected, which yields a contradiction. +z1 +i is identical to w1 +i or ˆw1 +i . This is an immediate contradiction since sources cannot have +degree 2 (recall that we already established Ra(i, 1) and Rb(i, 2) to be rays). +zi is incident to the first edges of one of the additional d1 +i outgoing paths. However, in +this case, Tcol can only be connected if there is precisely one further vertex of Tcol in V 1 +i +that is incident to all outgoing edges not covered by z1 +i . However, in this case, Tcol is an +invalid tree, yielding the desired contradiction. +Since the three cases above are exhaustive, the proof of Claim 5 is concluded. ■ +Next we need the following property: +Claim 6: Let t be a positive integer. If t < a then each ray in Rt +col is external. +For the proof, recall that |Rt| = |Rt +col| since T and Tcol are isomorphic. Note that each +ray R of length t of T is either fully contained in T ′, or it is one of the dj +i black rays for some +(i, j) ∈ [k] × [2]. (See Figure 12) If R is fully contained in T ′, then R is also contained in +Rt +col and it is external. +If R = r0, r1, . . . , rt is one of the dj +i black rays, then Tcol contains a path R′ = y0, r1, . . . , rt +for some y0 ∈ V j +i . Suppose that y0 has degree at least 3 in Tcol. Then, as in Claim 1, R′ is +then an external ray, and we are finished. We next consider the case where y0 has degree 1 +or 2 in Tcol. +If the degree is 1, then Tcol is disconnected, leading to a contradiction. If the degree is 2, +then y0 ̸= zj +i by Claim 5. Thus, the only way for Tcol not being disconnected is y0 = xj +i and +Tcol[Ea +i ] is a path. However, then we obtained a ray of length at least a + t which is neither +external, nor in the set ˆR. Thus, we obtain a contradiction by either Claim 2 (a + t < b), or +by Claim 1 (a + t = b), or by Lemma 73 (a + t > b). This concludes the proof of Claim 6. ■ + +L. A. Goldberg and M. Roth +51 +Next, observe that Tcol cannot connect z1 +i and z2 +i via a path within G, that is, via a path +containing the edges Ea +i : Otherwise Tcol would contain a cycle since p1 +i and p2 +i are connected +by a path within T ′. We will see that z1 +i and z2 +i are sources of Tcol. +Let S be the set of all sources of T. Consider the multi-set of leaf-degrees of T +degL(S) := {{degL(s) | s ∈ S}} . +Let Scol be the set of all sources of Tcol and let degL(Scol) be the muti-set of leaf-degrees Tcol. +Since Tcol and T are isomorphic, the multi-sets degL(S) and degL(Scol) are equal. +Suppose that s ∈ S is a source of T not among the designated sources sj +i. Then s is +contained in T ′, and it is also a source of Tcol. Since all of the zj +i have degree at least 3 in +Tcol (by Claim 5), they cannot be part of further rays with source s in Tcol so s has the same +leaf-degree in T and in Tcol. +We next show that for each i ∈ [k], the set V 1 +i ∪ V 2 +i contains at least 2 sources of Tcol: +Either z1 +i is a source or it is connected by a 2-path within Tcol[G] to another source. However, +the only vertices reachable in Tcol[G] from z1 +i that can have degree at least 3 are contained in +V 2 +i . Similarly, either x2 +i is a source or it is connected by a 2-path within Tcol[G] to a source +in V 1 +i . We have already seen that z1 +i cannot be connected to z2 +i within Tcol[G]. Thus the +sources reachable from z1 +i and z2 +i within Tcol[G] must be distinct, and we have shown that +for each i ∈ [k], the set V 1 +i ∪ V 2 +i contains at least 2 sources of Tcol. +Since Tcol and T have the same number of sources, and since 2k sources of T are not +contained in T ′, we have thus shown that for each i ∈ [k], the set V 1 +i ∪ V 2 +i contains precisely +2 sources of Tcol; let us denote those 2 sources by ˆz1 +i and ˆz2 +i . +Now, consider the following subsets of S and Scol: +S′ := {s1 +1, s2 +1, . . . , s1 +k, s2 +k} is the set of designated sources. +S′ +col := {ˆz1 +1, ˆz2 +1, . . . , ˆz1 +k, ˆz2 +k} is the set of sources of Tcol in G (within ˆG). +Since we already know that degL(S \ S′) = degL(Scol \ S′ +col) (those are the sources in T ′), we +require degL(S′) = degL(S′ +col) for T and Tcol to be isomorphic. +What follows is the final claim within the proof of this lemma. +Claim 7: For all i ∈ [k], the following five conditions are satisfied: +{z1 +i , z2 +i } = {ˆz1 +i , ˆz2 +i }, that is, z1 +i and z2 +i are the two sources in V 1 +i ∪ V 2 +i . +Tcol contains precisely 2 vertices in V 1 +i : One is z1 +i and one is x1 +i . +x1 +i has degree 1. Further, z1 +i , w1 +i , ˆw1 +i and all the endpoints of the d1 +i rays are the same. +Tcol contains precisely 1 vertex in V 2 +i . Further, z2 +i , x2 +i , w2 +i and all endpoints of the d2 +i +rays are the same. +Tcol[Ea +i ] is a ray with source z2 +i (= x2 +i = w2 +i ). +Before proving Claim 7, we point out that (1b) and (1f) follow immediately from Claim +7; see Figure 17 and observe that Tcol[Ea +i ] becomes the ray Ra(i, 2), and x1 +i becomes the +endpoint ua(i, 2) of Ra(i, 2) for each i ∈ [k]. Thus the proof of this lemma is concluded if +Claim 7 is proved, which is done below: +We first show that {z1 +i , z2 +i } = {ˆz1 +i , ˆz2 +i } for each i ∈ [k]. Let Φ = � +s∈S′ degL(s) and +Φcol = � +s∈S′ +col degL(s). Observe that degL(S′) = degL(S′ +col) implies Φ = Φcol. +We start by observing that +degL(ˆz1 +i ) + degL(ˆz2 +i ) ≤ (d1 +i + 2) + (d2 +i + 1) + 2. +There are d1 +i rays from V 1 +i and also Ra(i, 1) and Rb(i, 1). There are d2 +i rays from V 2 +i and +also Rb(i, 2). There is also Ea +i which could form two rays. + +52 +Parameterised and Fine-grained Subgraph Counting, modulo 2 +We next show that Ea +i cannot form two rays. Assume for contradiction that is does. +Since Tcol is connected, z1 +i , w1 +i , ˆw1 +i , x1 +i and all the endpoints of the d1 +i rays are identical, +and z2 +i , w2 +i , x2 +i and all the endpoints of the d2 +i rays are identical. +Now, if Tcol[Ea +i ] would be the disjoint union of two rays of length less than a with sources +z1 +i and z2 +i then those rays are non-external rays of length less than a, contradicting Claim +6. We have now shown +degL(ˆz1 +i ) + degL(ˆz2 +i ) ≤ (d1 +i + 2) + (d2 +i + 2) . +(11) +Next, note that by definition of the dj +i (see Figure 12), the following holds: +(d1 +i + 2) + (d2 +i + 2) = degL(s1 +i ) + degL(s2 +i ) +(12) +We have now shown that +degL(ˆz1 +i ) + degL(ˆz2 +i ) ≤ degL(s1 +i ) + degL(s2 +i ). +Finally, we will show that z1 +i and z2 +i are sources to finish the first bullet point. +Consider z1 +i , and recall that is has degree at least 3 by Claim 5, and assume for contradic- +tion that it is not a source of Tcol. Then z1 +i = x1 +i , and Tcol[Ea +i ] is a path, and x2 +i is source +(since it is the only vertex in V (Tcol) ∩ V 2 +i that might have degree at least 3, except for +z2 +i ). Note that this also implies that z2 +i is a source. Thus {ˆz1 +i , ˆz2 +i } = {x2 +i , z2 +i }. In this +case, we have +degL(ˆz1 +i ) + degL(ˆz2 +i ) ≤ d2 +i + 1 < degL(s1 +i ) + degL(s2 +i ) . +Consequently, using (11) and (12), we have Φcol < Φ, which is a contradiction. Thus z1 +i +is a source of Tcol, and a similar argument shows that z2 +i is a source of Tcol as well. +We now prove the remaining items. In what follows, using the previous bulleted item, we +can assume that w.l.o.g. ˆz1 +i = z1 +i and ˆz2 +i = z2 +i for all i ∈ [k]. First, recall that we ordered +the sj +i by their leaf-degrees, that is +degL(s1 +1) ≥ degL(s2 +1) ≥ · · · ≥ degL(s2 +k) ≥ 2 . +If x1 +1 were equal to z1 +1, then Tcol can only be connected if there is only one vertex in V 1 +1 , +that is, all edges incident to V 1 +1 are in fact incident to z1 +1. However, in that case, we have +degL(z1 +1) = degL(s1 +1) + 1 (by construction of ˆG), and thus the multi-sets cannot be equal +anymore. Hence x1 +1 ̸= z1 +1. +If x1 +1 had degree 2, then there would have been a ray of length at least a+1 that originates +in V 2 +1 (otherwise Tcol would have been disconnected). However, this ray would neither +be external, nor among the rays in ˆR, contradicting either Lemma 73 or the previous +sequence of claims. Finally, if x1 +1 had degree at least 3, then Tcol would have contained +more sources than T, which also yields a contradiction. +This shows that x1 +1 has degree 1. However, this implies that Tcol can only contain one vertex +in V 2 +i ; otherwise Tcol would be disconnected. Note that we have just proved the remaining +items of Claim 7 for i = 1. Additionally, we have shown that degL(z1 +1) = degL(s1 +1) and +degL(z2 +1) = degL(s2 +1) Hence we can remove those two numbers from the multi-sets and +continue recursively with i = 2. This concludes the proof of Claim 7, and thus the proof +of the overall lemma. +◀ +We are now ready to conclude the case for trees of unbounded fork number. + +L. A. Goldberg and M. Roth +53 +▶ Lemma 74. Let T be a recursively enumerable class of trees of unbounded fork number. +Then ⊕Sub(T ) is ⊕W[1]-hard. +Proof. We proceed similarly to Lemma 44. However, we have to take care of some subtleties. +First, we start with a class C of cubic bipartite graphs of unbounded treewidth. Next, we +wish to rely on Lemma 68 to obtain the identity +⊕Emb((Q +♯ +τ, cτ) → (G, c)) = ⊕ColSub(T → ( ˆG, ˆγ)), +where τ is the fracture defined in Definition 61. Unfortunately, Lemma 68 only yields the +above identity if, for each v ∈ V (Q), |c−1(v)| is odd, that is, each colour class of vertices of +G has odd cardinality. However, this property can easily be achieved. Let (G′, c′) be the +Q-coloured graph obtained from (G, c) by adding to each even colour class one fresh isolated +vertex. Since Q +♯ +τ does not have isolated vertices, this operation does not change the number +of colour-preserving embeddings. In combination with Lemma 68 we thus obtain +⊕Emb((Q +♯ +τ, cτ) → (G, c)) = ⊕Emb((Q +♯ +τ, cτ) → (G′, c′)) = ⊕ColSub(T → ( ˆG′, ˆγ)). +From here on, we can proceed analogously to the proof of Lemma 44. +◀ +4.4 +The Dichotomy Theorem for Trees +We are now able to prove Theorem 5, i.e., an exhaustive and explicit parameterised complexity +classification for counting trees modulo 2: +▶ Theorem 5. Let T be a recursively enumerable class of trees. If T is matching splittable, +then ⊕Sub(T ) is fixed-parameter tractable. Otherwise ⊕Sub(T ) is ⊕W[1]-complete. +Proof. The fixed-parameter tractability result, as well as the fact that ⊕Sub(T ) is always +contained in ⊕W[1] were both shown in [11]. Hence, it remains to prove ⊕W[1]-hardness if +T is not matching splittable. +By Lemma 32 each class T of trees that is not matching splittable has unbounded +C-number, unbounded star number, or unbounded fork number. Finally, each of these three +cases yields ⊕W[1]-hardness as established by Lemmas 44, 56, and 74. +◀ +References +1 +Noga Alon, Phuong Dao, Iman Hajirasouliha, Fereydoun Hormozdiari, and S. Cenk Sahinalp. +Biomolecular network motif counting and discovery by color coding. Bioinformatics, 24(13):i241– +i249, 2008. doi:10.1093/bioinformatics/btn163. +2 +Noga Alon, Raphael Yuster, and Uri Zwick. Color-coding. J. ACM, 42(4):844–856, 1995. +doi:10.1145/210332.210337. +3 +Suman K. Bera, Lior Gishboliner, Yevgeny Levanzov, C. Seshadhri, and Asaf Shapira. Counting +subgraphs in degenerate graphs. J. ACM, 69(3):23:1–23:21, 2022. doi:10.1145/3520240. +4 +Andreas Björklund, Holger Dell, and Thore Husfeldt. The parity of set systems under random +restrictions with applications to exponential time problems. +In Magnús M. Halldórsson, +Kazuo Iwama, Naoki Kobayashi, and Bettina Speckmann, editors, Automata, Languages, +and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, +2015, Proceedings, Part I, volume 9134 of Lecture Notes in Computer Science, pages 231–242. +Springer, 2015. doi:10.1007/978-3-662-47672-7\_19. +5 +Marco Bressan. Faster algorithms for counting subgraphs in sparse graphs. Algorithmica, +83(8):2578–2605, 2021. doi:10.1007/s00453-021-00811-0. + +54 +Parameterised and Fine-grained Subgraph Counting, modulo 2 +6 +Andrei A. Bulatov and Amirhossein Kazeminia. Complexity classification of counting graph +homomorphisms modulo a prime number. In Stefano Leonardi and Anupam Gupta, editors, +STOC ’22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, +June 20 - 24, 2022, pages 1024–1037. ACM, 2022. doi:10.1145/3519935.3520075. +7 +Jianer Chen, Benny Chor, Mike Fellows, Xiuzhen Huang, David W. Juedes, Iyad A. Kanj, +and Ge Xia. Tight lower bounds for certain parameterized NP-hard problems. Inf. Comput., +201(2):216–231, 2005. doi:10.1016/j.ic.2005.05.001. +8 +Jianer Chen, Xiuzhen Huang, Iyad A. Kanj, and Ge Xia. +Strong computational lower +bounds via parameterized complexity. J. Comput. Syst. Sci., 72(8):1346–1367, 2006. doi: +10.1016/j.jcss.2006.04.007. +9 +Yijia Chen, Marc Thurley, and Mark Weyer. Understanding the Complexity of Induced +Subgraph Isomorphisms. +In Proceedings of the 35th International Colloquium on Auto- +mata, Languages and Programming (ICALP), pages 587–596. Springer, 2008. doi:10.1007/ +978-3-540-70575-8\_48. +10 +Radu Curticapean. Counting matchings of size k is w[1]-hard. In Fedor V. Fomin, Rusins +Freivalds, Marta Z. Kwiatkowska, and David Peleg, editors, Automata, Languages, and +Programming - 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, +Proceedings, Part I, volume 7965 of Lecture Notes in Computer Science, pages 352–363. +Springer, 2013. doi:10.1007/978-3-642-39206-1\_30. +11 +Radu Curticapean, Holger Dell, and Thore Husfeldt. Modular counting of subgraphs: Match- +ings, matching-splittable graphs, and paths. In Petra Mutzel, Rasmus Pagh, and Grzegorz +Herman, editors, 29th Annual European Symposium on Algorithms, ESA 2021, September 6-8, +2021, Lisbon, Portugal (Virtual Conference), volume 204 of LIPIcs, pages 34:1–34:17. Schloss +Dagstuhl - Leibniz-Zentrum für Informatik, 2021. doi:10.4230/LIPIcs.ESA.2021.34. +12 +Radu Curticapean, Holger Dell, and Dániel Marx. Homomorphisms are a good basis for +counting small subgraphs. In Proceedings of the 49th Annual ACM SIGACT Symposium on +Theory of Computing (STOC), pages 210–223. ACM, 2017. doi:10.1145/3055399.3055502. +13 +Radu Curticapean and Dániel Marx. Complexity of counting subgraphs: Only the boundedness +of the vertex-cover number counts. In 55th IEEE Annual Symposium on Foundations of +Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 130–139. +IEEE Computer Society, 2014. doi:10.1109/FOCS.2014.22. +14 +Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin +Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. +doi:10.1007/978-3-319-21275-3. +15 +Víctor Dalmau and Peter Jonsson. The complexity of counting homomorphisms seen from the +other side. Theoret. Comput. Sci., 329(1-3):315–323, 2004. doi:10.1016/j.tcs.2004.08.008. +16 +Julian Dörfler, Marc Roth, Johannes Schmitt, and Philip Wellnitz. Counting induced subgraphs: +An algebraic approach to #w[1]-hardness. Algorithmica, 84(2):379–404, 2022. doi:10.1007/ +s00453-021-00894-9. +17 +Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. +Texts in Computer Science. Springer, 2013. doi:10.1007/978-1-4471-5559-1. +18 +Jörg Flum and Martin Grohe. The parameterized complexity of counting problems. SIAM J. +Comput., 33(4):892–922, 2004. doi:10.1137/S0097539703427203. +19 +Jörg Flum and Martin Grohe. +Parameterized Complexity Theory. +Texts in Theoretical +Computer Science. An EATCS Series. Springer, 2006. doi:10.1007/3-540-29953-X. +20 +Martin Grohe and Dániel Marx. On tree width, bramble size, and expansion. J. Comb. Theory, +Ser. B, 99(1):218–228, 2009. doi:10.1016/j.jctb.2008.06.004. +21 +Daniel J. Harvey and David R. Wood. The treewidth of line graphs. J. Comb. Theory, Ser. B, +132:157–179, 2018. doi:10.1016/j.jctb.2018.03.007. +22 +Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-sat. J. Comput. Syst. +Sci., 62(2):367–375, 2001. doi:10.1006/jcss.2000.1727. + +L. A. Goldberg and M. Roth +55 +23 +Bart M. P. Jansen and Dániel Marx. Characterizing the easy-to-find subgraphs from the +viewpoint of polynomial-time algorithms, kernels, and turing kernels. In Piotr Indyk, editor, +Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, +SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 616–629. SIAM, 2015. doi: +10.1137/1.9781611973730.42. +24 +Oleksii Kuchaiev, Tijana Milenković, Vesna Memišević, Wayne Hayes, and Nataša Pržulj. +Topological network alignment uncovers biological function and phylogeny. Journal of the +Royal Society Interface, 7(50):1341–1354, 2010. doi:10.1098/rsif.2010.0063. +25 +R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan, D. Chklovskii, and U. Alon. +Network +Motifs: Simple Building Blocks of Complex Networks. Science, 298(5594):824–827, 2002. +doi:10.1126/science.298.5594.824. +26 +Ron Milo, Shalev Itzkovitz, Nadav Kashtan, Reuven Levitt, Shai Shen-Orr, Inbal Ayzenshtat, +Michal Sheffer, and Uri Alon. Superfamilies of evolved and designed networks. Science, +303(5663):1538–1542, 2004. doi:10.1126/science.1089167. +27 +Norbert Peyerimhoff, Marc Roth, Johannes Schmitt, Jakob Stix, and Alina Vdovina. Paramet- +erized (modular) counting and cayley graph expanders. In Filippo Bonchi and Simon J. Puglisi, +editors, 46th International Symposium on Mathematical Foundations of Computer Science, +MFCS 2021, August 23-27, 2021, Tallinn, Estonia, volume 202 of LIPIcs, pages 84:1–84:15. +Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. doi:10.4230/LIPIcs.MFCS.2021.84. +28 +Norbert Peyerimhoff, Marc Roth, Johannes Schmitt, Jakob Stix, and Alina Vdovina. Paramet- +erized (modular) counting and cayley graph expanders. CoRR, abs/2104.14596, 2021. URL: +https://arxiv.org/abs/2104.14596, arXiv:2104.14596. +29 +Norbert Peyerimhoff, Marc Roth, Johannes Schmitt, Jakob Stix, Alina Vdovina, and Philip +Wellnitz. Parameterized Counting and Cayley Graph Expanders. SIAM J. Discrete Math., to +appear. +30 +Marc Roth, Johannes Schmitt, and Philip Wellnitz. Detecting and Counting Small Subgraphs, +and Evaluating a Parameterized Tutte Polynomial: Lower Bounds via Toroidal Grids and +Cayley Graph Expanders. CoRR, abs/2011.03433, 2020. arXiv:2011.03433. +31 +Marc Roth, Johannes Schmitt, and Philip Wellnitz. Detecting and Counting Small Subgraphs, +and Evaluating a Parameterized Tutte Polynomial: Lower Bounds via Toroidal Grids and +Cayley Graph Expanders. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, +48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July +12-16, 2021, Glasgow, Scotland (Virtual Conference), volume 198 of LIPIcs, pages 108:1–108:16. +Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. doi:10.4230/LIPIcs.ICALP.2021. +108. +32 +Benjamin Schiller, Sven Jager, Kay Hamacher, and Thorsten Strufe. StreaM – A Stream-Based +Algorithm for Counting Motifs in Dynamic Graphs. In Proceedings of the 2nd International +Conference on Algorithms for Computational Biology (AlCoB), pages 53–67. Springer Interna- +tional Publishing, 2015. doi:10.1007/978-3-319-21233-3_5. +33 +Seinosuke Toda. PP is as hard as the polynomial-time hierarchy. SIAM J. Comput., 20(5):865– +877, 1991. doi:10.1137/0220053. +34 +Ngoc Hieu Tran, Kwok Pui Choi, and Louxin Zhang. Counting motifs in the human interactome. +Nature communications, 4(1):1–8, 2013. doi:10.1038/ncomms3241. +35 +Charalampos E. Tsourakakis, Jakub Pachocki, and Michael Mitzenmacher. Scalable motif- +aware graph clustering. In Proceedings of the 26th International Conference on World Wide +Web (WWW), page 1451–1460, 2017. doi:10.1145/3038912.3052653. +36 +Virginia Vassilevska Williams, Joshua R. Wang, Richard Ryan Williams, and Huacheng Yu. +Finding four-node subgraphs in triangle time. In Piotr Indyk, editor, Proceedings of the Twenty- +Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, +USA, January 4-6, 2015, pages 1671–1680. SIAM, 2015. doi:10.1137/1.9781611973730.111. + diff --git a/4dAzT4oBgHgl3EQfuv2o/content/tmp_files/load_file.txt b/4dAzT4oBgHgl3EQfuv2o/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..88ebc718c97583133ab434a51bbab2ebef8ab81c --- /dev/null +++ b/4dAzT4oBgHgl3EQfuv2o/content/tmp_files/load_file.txt @@ -0,0 +1,2181 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf,len=2180 +page_content='Parameterised and Fine-grained Subgraph Counting, modulo 2∗ Leslie Ann Goldberg University of Oxford Marc Roth University of Oxford Abstract Given a class of graphs H, the problem ⊕Sub(H) is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The input is a graph H ∈ H together with an arbitrary graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The problem is to compute, modulo 2, the number of subgraphs of G that are isomorphic to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The goal of this research is to determine for which classes H the problem ⊕Sub(H) is fixed-parameter tractable (FPT), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=', solvable in time f(|H|) · |G|O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Curticapean, Dell, and Husfeldt (ESA 2021) conjectured that ⊕Sub(H) is FPT if and only if the class of allowed patterns H is matching splittable, which means that for some fixed B, every H ∈ H can be turned into a matching (a graph in which every vertex has degree at most 1) by removing at most B vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Assuming the randomised Exponential Time Hypothesis, we prove their conjecture for (I) all hereditary pattern classes H, and (II) all tree pattern classes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=', all classes H such that every H ∈ H is a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We also establish almost tight fine-grained upper and lower bounds for the case of hereditary patterns (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 2012 ACM Subject Classification Theory of computation → Problems, reductions and completeness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Mathematics of computing → Discrete mathematics Keywords and phrases modular counting, parameterised complexity, fine-grained complexity, sub- graph counting Acknowledgements We want to thank Radu Curticapean, Holger Dell and Thore Husfeldt for insightful discussions on an early draft of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 1 Introduction The last two decades have seen remarkable progress in the classification of subgraph counting problems: Given a small pattern graph H and a large host graph G, how often does H occur as a subgraph if G?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since it was discovered that subgraph counts from small patterns reveal global properties of complex networks [25, 26], subgraph counting has also found several applications in fields such as biology [1, 32] genetics [34], phylogeny [24], and data mining [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Moreover, the theoretical study of subgraph counting and related problems has led to many deep structural insights, establishing both new algorithmic techniques and tight lower bounds under the lenses of fine-grained and parameterised complexity theory [18, 15, 9, 13, 12, 5, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Without any additional restrictions, the subgraph counting problem is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The complexity class #W[1] is the parameterised complexity class analgous to NP (see Section 2 for more detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Under standard assumptions, problems that are #W[1]-hard are not fixed- parameter tractable (FPT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' However, the canonical complete problem for #W[1], the problem of counting k-cliques, corresponds to the special case of the subgraph counting problem ∗ For the purpose of Open Access, the authors have applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' All data is provided in full in the results section of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='01696v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='CC] 4 Jan 2023 2 Parameterised and Fine-grained Subgraph Counting, modulo 2 where H is a clique of size k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This problem cannot be solved in time f(k) · no(k) for any function f unless the Exponential Time Hypothesis (ETH) fails [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Due to this hardness result, the research focus in this area shifted to the question: Under which restrictions on the patterns H and the hosts G is algorithmic progress possible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' More precisely, under which restrictions can the problem be solved in time f(|H|) · |G|O(1), for some computable function f?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Instances that can be solved within such a run time bound are called fixed-parameter tractable (FPT);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' allowing a potential super-polynomial overhead in the size of the pattern |H| formalises the assumption that H is assumed to be (significantly) smaller than G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If only the patterns are restricted, then the situation if fully understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Formally, given a class H of patterns, the problem #Sub(H) asks, given as input a graph H ∈ H and an arbitrary graph G, to compute the number of subgraphs of G that are isomorphic to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Following initial work by Flum and Grohe [18] and by Curticapean [10], Curticapean and Marx [13] proved that, under standard assumptions, #Sub(H) is FPT if and only if H has bounded matching number, that is, if there is a positive integer B such that the size of any matching in any graph in H is at most B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' They also proved that all FPT cases are polynomial-time solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In stark contrast, almost nothing is known for the decision version Sub(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Here, the task is to correctly decide whether there is a copy of H ∈ H in G, rather than to count the copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' It is known that Sub(H) is FPT whenever H has bounded treewidth (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' [19, Chapter 13]), and it is conjectured that those are all FPT cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' However, resolving this conjecture belongs to the “most infamous” open problems in parameterised complexity theory [17, Chapter 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1 Counting Modulo 2 To interpolate between the fully understood realm of (exact) counting and the barely understood realm of decision, Curticapean, Dell and Husfeldt proposed the study of counting subgraphs, modulo 2 [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Formally, they introduced the problem ⊕Sub(H), which expects as input a graph H ∈ H and an arbitrary graph G, and the goal is to compute modulo 2 the number of subgraphs of G isomorphic to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The study of counting modulo 2 received significant attention from the viewpoint of classical and structural complexity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For example, one way to state Toda’s Theorem [33] is PH ⊆ P⊕P, implying that counting satisfying assignments of a CNF, modulo 2, is at least as hard as the polynomial hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Another example is the quest to classify the complexity of counting modulo 2 the homomorphisms to a fixed graph, which was very recently resolved by Bulatov and Kazeminia [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In their work [11], Curticapean, Dell and Husfeldt proved that the problem of counting k-matchings modulo 2, that is, the problem ⊕Sub(H) where H is the class of all 1-regular graphs, is fixed-parameter tractable, where the parameter k is |H|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since the exact counting version of this problem is #W[1]-hard [10], their result provides an example where counting modulo 2 is strictly easier than exact counting (subject to complexity assumptions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The complexity class ⊕W[1] can be defined via the complete problem of counting k-cliques modulo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Crucially, ⊕W[1]-hard problems are not fixed-parameter tractable, unless the randomised ETH (rETH) fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Curticapean et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' [11] proved that counting k-paths modulo 2 is ⊕W[1]-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since finding a k-path in a graph G is fixed-parameter tractable via colour- coding [2], this hardness result provides an example where counting modulo 2 is strictly harder than decision (subject to complexity assumptions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Combining those observations, it appears that counting subgraphs modulo 2 may lie strictly in between the complexity of decision and the complexity of exact counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Goldberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Roth 3 A matching is a graph whose degree is at most 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The matching-split number of a graph H is the minimum size of a set S ⊆ V (H) such that H \\ S is a matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A class of graphs H is called matching splittable if there is a positive integer B such that the matching-split number of any H ∈ H is at most B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For example, the class of all matchings is matching splittable while the class of all cycles is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Curticapean, Dell and Husfeldt extended their FTP algorithm for counting k-matchings modulo 2 to obtain an FPT algorithm for ⊕Sub(H) for any matching-splittable class H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' On this basis, they then made the following conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Conjecture 1 ([11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ⊕Sub(H) is FPT if and only if H is matching splittable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A class H of graphs is called hereditary if it is closed under vertex removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Intriguingly, if Conjecture 1 is true, then the FPT criterion for counting subgraphs modulo 2 (⊕Sub(H)) would coincide with the polynomial-time criterion for finding subgraphs (Sub(H)) for hered- itary pattern classes H as established by Jansen and Marx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Theorem 2 ([23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let H be a hereditary class of graphs and assume P ̸= NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then Sub(H) is solvable in polynomial time if and only if H is matching splittable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Jansen and Marx also conjecture that the condition of H being hereditary can be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Conjecture 3 ([23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Sub(H) is solvable in polynomial time if and only if H is matching splittable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Conjectures 1 and 3 have the remarkable consequence that ⊕Sub(H) is FPT if and only if Sub(H) is solvable in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In the current work we establish this consequence for all hereditary pattern classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='2 Our Contributions We resolve Conjecture 1 for all hereditary classes H, as well as for every class H consisting only of trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let H be a hereditary class of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If H is matching splittable, then ⊕Sub(H) is fixed-parameter tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Otherwise, the problem is ⊕W[1]-complete and, assuming rETH, cannot be solved in time f(|H|) · |G|o(|V (H)|/ log |V (H)|) for any function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let T be a recursively enumerable class of trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If T is matching splittable, then ⊕Sub(T ) is fixed-parameter tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Otherwise ⊕Sub(T ) is ⊕W[1]-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In order to prove our classifications, we adapt the by-now-standard technique for ana- lysing subgraph counting problems established by Curticapean, Dell and Marx [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let #Sub(H → G) denote the number of subgraphs of a graph G that are isomorphic to a graph H and let #Hom(F → G) denotes the number of homomorphisms (edge-preserving mappings) from a graph F to a graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Given a graph H, there is a function aH from graphs to rationals with finite support such that the following holds for any graph G: #Sub(H → G) = � F aH(F) · #Hom(F → G) , (1) where the sum is over all (isomorphism types of) graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since aH has finite support, aH(F) = 0 for all but finitely-many graphs F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus, equation (1) allows us to express the solution to the exact counting problem as a finite linear combination of homomorphism counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In a nutshell, the framework of [12] states that computing the function G �→ #Sub(H → G) 4 Parameterised and Fine-grained Subgraph Counting, modulo 2 is hard to compute if and only if there is a graph F of high treewidth with aH(F) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This translates the complexity of (exact) subgraph counting to the purely combinatorial problem of understanding the coefficients aH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' One might hope that this strategy transfers to counting modulo 2 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Unfortunately, this is not possible as Equation (1) might not be well-defined if arithmetic is done modulo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The reason for this is the fact that the coefficients aH(F) are of the form µ(F, H) × |Aut(H)|−1, where µ(F, H) is an integer, and Aut(H) is the automorphism group of the graph H [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus there is, a priori, no hope to extend the framework to counting modulo 2 for pattern graphs with an even number of automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In fact, according to Curticapean, Dell and Husfeldt [11], the absence of a comparable framework for counting modulo 2 is one of the main challenges for establishing the hardness part of Conjecture 1, and it is the main reason why the reductions in [11] use more classical, gadget-based reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In this work, we solve the problem of patterns with an even number of automorphisms by considering a colourful intermediate problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' More concretely, we will equip each edge of the pattern H with a distinct colour and show that it will be sufficient to consider only automorphisms that preserve the colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If H has no isolated vertices, then this is only the trivial automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Formally, the coloured approach will be based on the notion of so-called fractured graphs introduced by Peyerimhoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In what follows (Section 2), we will first introduced all required notions and previous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In Section 3, we will prove the classification for hereditary pattern classes (Theorem 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' On a technical level, this proof can be considered a warm-up for the significantly harder challenge of establishing the classification for trees (Theorem 5), which we prove in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 2 Preliminaries Let f : A1 × A2 → B be a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each a1 ∈ A1 we write f(a1, ⋆) : A2 → B for the function that maps a2 ∈ A2 to f(a1, a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Graphs in this work are undirected and without self loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A homomorphism from a graph H to a graph G is a mapping ϕ from the vertices V (H) of H to the vertices V (G) of G such that for each edge e = {u, v} ∈ E(H) of H, the image ϕ(e) = {ϕ(u), ϕ(v)} is an edge of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A homomorphism is called an embedding if it is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We write Hom(H → G) and Emb(H → G) for the sets of homomorphisms and embeddings, respectively, from H to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' An embedding ϕ ∈ Emb(H → G) is called an isomorphism if it is bijective and {u, v} ∈ E(H) ⇔ {ϕ(u), ϕ(v)} ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We say that H and G are isomorphic, denoted by H ∼= G, if an isomorphism from H to G exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A graph invariant ι is a function from graphs to rationals such that ι(H) = ι(G) for each pair of isomorphic graphs H and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A subgraph of G is a graph G′ with V (G′) ⊆ V (G) and E(G′) ⊆ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We write Sub(H → G) for the set of all subgraphs of G that are isomorphic to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Given a subset of vertices S ⊆ V (G) of a graph G, we write G[S] for the graph induced by S, that is, G[S] has vertices S and edges {{u, v} ⊆ S | {u, v} ∈ E(G)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We denote by tw(G) the treewidth of the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since we will rely on treewidth purely in a black-box manner, we omit the technical definition and refer the reader to [14, Chapter 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Given any graph invariant ι (such as treewidth) and a class of graphs G, we say that ι is bounded in G if there is a non-negative integer B such that, for all G ∈ G, ι(G) ≤ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Otherwise we say that ι is unbounded in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Given a graph H = (V, E), a splitting set of H is a subset of vertices S such that every vertex in H[V \\S] has degree at most 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The matching-split number of H is the minimum size L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Goldberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Roth 5 v vB1 vB2 Figure 1 Illustration of the construction of a fractured graph from [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The left picture shows a vertex v of a graph Q with incident edges EQ(v) = { , , , , , }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The right picture shows the splitting of v in the construction of the fractured graph Q ♯ σ for a fracture σ satisfying that the partition σv contains two blocks B1 = { , , }, and B2 = { , , }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' of a splitting set of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A class of graphs H is called matching splittable the matching-split number of H is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1 Colour-Preserving Homomorphisms and Embeddings A homomorphism c from a graph G to a graph Q is sometimes called a “Q-colouring” of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A Q-coloured graph is a pair consisting of a graph G and a homomorphism c from G to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that the identity function idQ on V (Q) is a Q-colouring of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If a homomorphism c from G to Q is vertex surjective, then we call (G, c) a surjectively Q-coloured graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Definition 6 (cE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A Q-colouring c of a graph G induces a (not necessarily proper) edge-colouring cE : E(G) → E(Q) given by cE({u, v}) = {c(u), c(v)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Notation: Given a Q-coloured graph (G, c) and a vertex u ∈ V (Q), we will use the capitalised letter U to denote the subset of vertices of G that are coloured by c with u, that is, U := c−1(u) ⊆ V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Given two Q-coloured graphs (H, cH) and (G, cG), we call a homomorphism ϕ from H to G colour-preserving if for each v ∈ V (H) we have cG(ϕ(v)) = cH(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We note the special case in which Q = H and cH is the identity idQ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' then the condition simplifies to cG(ϕ(v)) = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A colour-preserving embedding of (H, cH) in (G, cG) is a vertex injective colour- preserving homomorphism from (H, cH) to (G, cG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We write Hom((H, cH) → (G, cG)) and Emb((H, cH) → (G, cG)) for the sets of all colour-preserving homomorphisms and embeddings, respectively, from (H, cH) to (G, cG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let k be a positive integer, let H be a graph with k edges, and let (G, γ) be a pair consisting of a graph G and a function that maps each edge of G to one of k distinct colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We refer to γ as a “k-edge colouring” of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For example, in most of our applications we will fix a graph Q with k edges and a Q-colouring c of G and we will take γ to be the edge-colouring cE from Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We write ColSub(H → (G, γ)) for the set of all subgraphs of G that are isomorphic to H and that contain each of the k edge colours precisely once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='2 Fractures and Fractured Graphs In this work, we will crucially rely on and extend the framework of fractured graphs as introduced in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Definition 7 (Fractures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let Q be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each vertex v of Q, let EQ(v) be the set of edges of Q that are incident to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A fracture of Q is a tuple ρ = (ρv)v∈V (Q), where for each vertex v of Q, ρv is a partition of EQ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 6 Parameterised and Fine-grained Subgraph Counting, modulo 2 Note that a fracture describes how to split (or how to fracture) each vertex of a given graph: for each vertex v, create a vertex vB for each block B in the partition ρv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' edges originally incident to v are made incident to vB if and only if they are contained in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We call the resulting graph the fractured graph H ♯ ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' a formal definition is given in Definition 8, a visualisation is given in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Definition 8 (Fractured Graph Q ♯ ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Given a graph Q, we consider the matching MQ containing one edge for each edge of Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' formally, V (MQ) := � e={u,v}∈E(Q) {ue, ve} and E(MQ) := {{ue, ve} | e = {u, v} ∈ E(Q)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For a fracture ρ of Q, we define the graph Q ♯ ρ to be the quotient graph of MQ under the equivalence relation on V (MQ) which identifies two vertices ve, wf of MQ if and only if v = w and e, f are in the same block B of the partition ρv of EQ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We write vB for the vertex of Q ♯ ρ given by the equivalence class of the vertices ve (for which e ∈ B) of MQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Definition 9 (Canonical Q-colouring cρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let Q be a graph and let ρ be a fracture of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The canonical Q-colouring of the fractured graph Q ♯ ρ maps vB to v for each v ∈ V (Q) and block B ∈ ρv, and is denoted by cρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Observe that cρ is the identity in V (Q) if ρ is the coarsest fracture (that is, each partition ρv only contains one block, in which case Q ♯ ρ = Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='3 Parameterised and Fine-grained Computation A parameterised computational problem is a pair consisting of a function P : Σ∗ → {0, 1} and a computable parameterisation κ : Σ∗ → N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A fixed-parameter tractable (FPT) algorithm for (P, κ) is an algorithm that computes P and runs, on input x ∈ Σ∗, in time f(κ(x)) · |x|O(1) for some computable function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We call (P, κ) fixed-parameter tractable (FPT) if an FPT algorithm for (P, κ) exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A parameterised Turing-reduction from (P, κ) to (P ′, κ′) is an FPT algorithm for (P, κ) that is equipped with oracle access to P ′ and for which there is a computable function g such that, on input x, each oracle query y satisfies κ′(y) ≤ g(κ(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We write (P, κ) ≤fpt T (P ′, κ′) if a parameterised Turing-reduction from (P, κ) to (P ′, κ′) exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This guarantees that fixed-parameter tractability of (P ′, κ′) implies fixed-parameter tractability of (P, κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For a more comprehensive introduction, we refer the reader the standard textbooks [14] and [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Counting modulo 2 and the rETH The lower bounds in this work will rely on the hardness of the parameterised complexity class ⊕W[1], which can be considered a parameterised equivalent of ⊕P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Following [11], we define ⊕W[1] via the complete problem ⊕Clique: Given as input a graph G and a positive integer k, the goal is to compute the number of k-cliques in G modulo 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=', to compute ⊕Sub(Kk → G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The problem is parameterised by k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A parameterised problem (P, κ) is called ⊕W[1]-hard if ⊕Clique ≤fpt T (P, κ), and it is called ⊕W[1]-complete if, additionally, (P, κ) ≤fpt T ⊕Clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Modifications of the classical Isolation Lemma (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' [4] and [36]) yield a randomised parameterised Turing reduction from finding a k-clique to computing the parity of the number of k-cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In combination with existing fine-grained lower bounds for finding a k-clique [7, 8], it can then be shown that ⊕Clique cannot be solved in time f(k) · |G|o(k) for any function f, unless the randomised Exponential Time Hypothesis fails: L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Goldberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Roth 7 ▶ Definition 10 (rETH, [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The randomised Exponential Time Hypothesis (rETH) asserts that 3-SAT cannot be solved by a randomised algorithm in time exp o(n), where n is the number of variables of the input formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' As an immediate consequence, the rETH implies that ⊕W[1]-hard problems are not fixed- parameter tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For the lower bounds in this work, we won’t reduce from ⊕Clique directly, but instead from the following, more general problem: ▶ Definition 11 (⊕cp-Hom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let H be a class of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The problem ⊕cp-Hom(H) has as input a graph H ∈ H and a surjectively H-coloured graph (G, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The goal is to compute ⊕Hom((H, idH) → (G, c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The problem is parameterised by |H|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The following lower bound was proved independently in [27, 29] and [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let H be a recursively enumerable class of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If the treewidth of H is unbounded then ⊕cp-Hom(H) is ⊕W[1]-hard and, assuming the rETH, it cannot be solved in time f(|H|) · |G|o(tw(H)/ log tw(H)) for any function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Next is the central problem in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Definition 13 (⊕Sub).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let H be a class of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The problem ⊕Sub(H) has as input a graph H ∈ H and a graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The goal is to compute ⊕Sub(H → G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The problem is parameterised by |H|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For example, writing K for the set of all complete graphs, the problem ⊕Sub(K) is equivalent to ⊕Clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Complexity Monotonicity and Inclusion-Exclusion Throughout this work, we will rely on two important tools introduced in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For the sake of being self-contained, we encapsulate them below in individual lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The first tool is an adaptation of the so-called Complexity Monotonicity principle to the realm of fractured graphs and modular counting (see [29, Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='3] for a detailed treatment and for a proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Intuitively, the subsequent lemma states that evaluating, modulo 2, a linear combination of colour-prescribed homomorphism counts from fractured graphs, is as hard as evaluating its hardest term with an odd coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Lemma 14 ([29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' There is a deterministic algorithm A and a computable function f such that the following conditions are satisfied: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A expects as input a graph Q and a Q-coloured graph (G, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A is equipped with oracle access to a function (G′, c′) �→ � ρ a(ρ) · ⊕Hom((Q ♯ ρ, cρ) → (G′, c′)) mod 2 , where the sum is over all fractures of Q and a is a function from fractures of Q to integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Each oracle query (G′, c′) is of size at most f(|Q|) · |G|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A computes ⊕Hom((Q ♯ ρ, cρ) → (G, c)) for each fracture ρ with a(ρ) ̸= 0 mod 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The running time of A is bounded by f(|Q|) · |G|O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The second tool is a standard application of the inclusion-exclusion principle (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' [29, Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='2 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' It will be used in the final steps of our reductions to remove the colourings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 8 Parameterised and Fine-grained Subgraph Counting, modulo 2 ▶ Lemma 15 ([29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' There is a deterministic algorithm A that satisfies the following condi- tions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A expects as input a graph H with k edges, a graph G and a k-edge colouring γ of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A is equipped with oracle access to the function ⊕Sub(H → ⋆), and each oracle query G′ satisfies |G′| ≤ |G|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A computes ⊕ColSub(H → (G, γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The running time of A is bounded by 2|H| · |G|O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 3 Classification for Hereditary Graph Classes In this section, we will completely classify the complexity of ⊕Sub(H) for hereditary classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let us start by restating the classification theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let H be a hereditary class of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If H is matching splittable, then ⊕Sub(H) is fixed-parameter tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Otherwise, the problem is ⊕W[1]-complete and, assuming rETH, cannot be solved in time f(|H|) · |G|o(|V (H)|/ log |V (H)|) for any function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The proof of Theorem 4 is split in four cases, which stem from a structural property of non matching splittable hereditary graph classes H due to Jansen and Marx [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For the statement, we need to consider the following classes: Fω is the class of all complete graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Fβ is the class of all complete bipartite graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' FP2 is the class of all P2-packings, that is, disjoint unions of paths with two edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1 FK3 is the class of all triangle packings, that is, disjoint unions of the complete graph of size 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Theorem 16 (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='5 in [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let H be a hereditary class of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If H is not matching splittable then at least one of the following are true: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=') Fω ⊆ H, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=') Fβ ⊆ H, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=') FP2 ⊆ H, or (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=') FK3 ⊆ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus, it suffices to consider cases 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We start with the easy cases of cliques and bicliques;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' they follow implicitly from previous works [11, 16, 27] and we only include a proof for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that a tight bound under rETH is known for those cases: ▶ Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let H be a hereditary class of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If Fω ⊆ H or Fβ ⊆ H then ⊕Sub(H) is ⊕W[1]-hard and, assuming rETH, cannot be solved in time f(|H|) · |G|o(|V (H)|) for any function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If Fω ⊆ H then ⊕W[1]-hardness follows immediately from the fact that ⊕Clique is the canonical ⊕W[1]-complete problem [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For the rETH lower bound, we can reduce from the problem of deciding the existence of a k-clique via a (randomised) reduction using a version of the Isolation Lemma due to Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' [36, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This reduction does not increase k or the size of the host graph and is thus tight with respect to the well-known lower bound for the clique problem due to Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' [7, 8]: Deciding the existence of a k-clique in an n-vertex graph cannot be done in time f(k) · no(k) for any function f, unless ETH fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Our lower bound under rETH follows since the reduction is randomised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 1 To avoid confusion, we remark that [23] uses P3 to denote the path of two edges (and three vertices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In the current work, it will be more convenient to use the number of edges of a path as index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Goldberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Roth 9 If Fβ ⊆ H, then the claim holds by [16, Theorem 5], which established the problem of counting, modulo 2, the induced copies of a k-by-k-biclique in an n-vertex bipartite graph to be ⊕W[1]-hard and not solvable in time f(k) · no(k) for any function f, unless rETH fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since a copy of a biclique (with at least one edge) in a bipartite graph must always be induced, the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This concludes the proof of Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ◀ The more interesting cases are FP2 ⊆ H and FK3 ⊆ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' One reason for this is that, in contrast to cliques and bicliques, the decision version of those instances are fixed-parameter tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Hence a reduction from the decision version via e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' an isolation lemma does not help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In other words, establishing hardness for those cases requires us to rely on the full power of counting modulo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' More precisely, we will rely on the framework of fractures graphs (see Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Both cases can be considered simpler applications of the machinery used in the later sections, so we will present all steps in great detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' While this might seem unnecessary given the simplicity of the constructions, we hope that it enables the reader to make themselves familiar with the general reduction strategies which will be used throughout the later sections of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1 Triangle Packings The goal of this subsection is to establish hardness of ⊕Sub(FK3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' To this end, let ∆ be an infinite computable class of cubic bipartite expander graphs, and let Q = {L(H) | H ∈ ∆} where L(H) is constructed as follows: Each v ∈ V (H) becomes a triangle with vertices vx, vy, and vz corresponding to the three neighbours x, y, and z of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Finally, for every edge {u, v} ∈ E(H) we identify vu and uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In fact, L(H) is just the line graph of H: Every edge of H becomes a vertex in L(H), and two vertices of L(H) are made adjacent if and only if the corresponding edges in H are incident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since all H ∈ ∆ are bipartite (and thus triangle-free), we can easily observe the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='2 ▶ Observation 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The mapping v �→ (vx, vy, vz) is a bijection from vertices of H to triangles in L(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We also consider the fracture of L(H) that splits L(H) back into |V (H)| triangles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' consider Figure 2 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Definition 19 (τ(H)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let H ∈ ∆ and recall that each vertex w of L(H) is obtained by identifying vu and uv for some edge {u, v} ∈ E(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Moreover, w has four incident edges ex, ey, ea, eb, to vx, vy, ua, ub, respectively, where x, y, u are the neighbours of v in H and v, a, b are the neighbours of u in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We define τ(H)w := {{ex, ey}, {ea, eb}}, and we proceed similar for all vertices of L(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Next, we use that tw(L(H)) = Ω(tw(H)) (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Moreover, tw(L(H)) ≤ |V (L(H))| since the treewidth of a graph is always bounded by the number of its vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Additionally, |V (L(H))| = |E(H)| by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since the graphs in ∆ are cubic, we further have that |E(H)| = Θ(|V (H)|) for H ∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We combine those bounds with the fact that expander graphs have treewidth linear in the number of vertices (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' [20]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' therefore ∆ and thus Q have unbounded treewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Putting these facts together, we obtain the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Fact 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Q has unbounded treewidth and tw(L(H)) = Θ(|V (L(H))|) = Θ(|V (H)|) for H ∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 2 Observation 18 is also an immediate consequence of Whitney’s Isomorphism Theorem implying that a triangle of a line graph corresponds to either a claw or to a triangle in its primal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 10 Parameterised and Fine-grained Subgraph Counting, modulo 2 We are now able to establish hardness of ⊕Sub(FK3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The proof will heavily rely on the transformation from edge-coloured subgraphs to homomorphisms established in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The problem ⊕Sub(FK3) is ⊕W[1]-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Furthermore, on input kK3 and G, the problem cannot be solved in time f(k) · |G|o(k/ log k) for any function f, unless rETH fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We reduce from ⊕cp-Hom(Q), which, by Fact 20 and Theorem 12, is ⊕W[1]-hard and for L(H) ∈ Q, it cannot be solved in time f(|L(H)|) · |G|o(|V (L(H))|/ log |V (L(H))|), unless rETH fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let L and (G, c) be an input instance to ⊕cp-Hom(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Recall that ∆ is computable — that is, there is an algorithm that takes a graph H and determines whether it is in ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus, there is an algorithm that takes input L ∈ Q and finds a graph H ∈ ∆ with L = L(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The run time of this algorithm depends on |L| but clearly not on (G, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let k = |V (H)| and note that |E(L(H))| = 3k, since, by construction, each vertex v of H becomes a triangle of L(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We consider the graph G as a 3k-edge-coloured graph, coloured by cE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' That is, each edge e = {x, y} of G is assigned the colour cE(e) = {c(x), c(y)} which is an edge of L (see Figure 2 for an illustration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Now, for any L-coloured graph (G′, c′) recall that ColSub(kK3 → (G′, c′ E)) is the set of subgraphs of G′ that are isomorphic to kK3 and that include each edge colour (each edge of L) precisely once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We will see later that ⊕ColSub(kK3 → (G′, c′ E)) can be computed using our oracle for ⊕Sub(FK3) using the principle of inclusion and exclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' It was shown in [29, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1] that there is a unique function a such that for every L-coloured graph (G′, c′) we have3 #ColSub(kK3 → (G′, c′ E)) = � ρ a(ρ) · Hom(L ♯ ρ → (G′, c′)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (2) where the sum is over all fractures of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Additionally, it was shown in [29, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='3] that a(⊤) = � ρ∈F(kK3,L) � w∈V (L) (−1)|ρw|−1 · (|ρw| − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , (3) where ⊤ is the fracture in which each partition consists only of one block (that is, L ♯ ⊤ = L), and F(kK3, L) is the set of all fractures ρ of L such that L ♯ ρ ∼= kK3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' However, note that, by Observation 18, there is only way to fracture L into k disjoint triangles, and this fracture is given by τ(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus, (3) simplifies to a(⊤) = � w∈V (L) (−1)|τ(H)w|−1 · (|τ(H)w| − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , (4) which is odd since each partition of τ(H) consists of precisely two blocks (so in fact the expression in (4) is (−1)|V (L)|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that the algorithm for ⊕cp-Hom(Q) is supposed to compute ⊕Hom((L, idL) → (G, c)) which is equal to ⊕Hom(L ♯ ⊤ → (G, c⊤)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since a(⊤) is odd, we can invoke Lemma 14 to recover this term by evaluating the entire linear combination (2), that is, by evaluating the function ⊕ColSub(kK3 → ⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' More concretely, this means that we need to compute ⊕ColSub(kK3 → (G′, c′ E)) for some L-coloured graphs (G′, c′) of size at most f(|L|) · |G| for 3 In the language of [29], Equation (2) is obtained by choosing Φ as the property of being isomorphic to kK3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Goldberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Roth 11 Figure 2 (Top:) A cubic bipartite graph H ∈ ∆, its line graph L(H), and the fractured graph induced by τ(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (Below:) An L(H)-coloured graph (G, c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' emphasised in distinct colours is the edge-colouring cE of G induced by the mapping {u, v} �→ {c(u), c(v)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Additionally we depict an element S ∈ ColSub(kK3 → (G, cE)), that is, a subgraph of G isomorphic to kK3 that contains each edge colour of G precisely once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 12 Parameterised and Fine-grained Subgraph Counting, modulo 2 some computable function f (see 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' in Lemma 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This can easily be done using Lemma 15 since we have oracle access to the function ⊕Sub(kK3 → ⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We emphasise that, by condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' of Lemma 15, each oracle query ˆG satisfies | ˆG| ≤ |G′|, where (G′, c′) is the L-coloured graph for which we wish to compute ⊕ColSub(kK3 → (G′, c′ E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since |(G′, c′)| ≤ f(|L|) · |G|, we obtain that | ˆG| ≤ f(|L|) · |G| as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since, by Fact 20, k = Θ(|kK3|) = Θ(|V (L)|) = Θ(tw(L)), our reduction yields ⊕W[1]- hardness and transfers the conditional lower bound under rETH as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ◀ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='2 P2-packings Next we establish hardness for the case of P2-packings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The strategy will be similar in spirit to the construction for triangle packings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' however, rather then identifying a unique fracture for which the technique applies, we will encounter an odd number of possible fractures in the current section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let ∆ be a computable infinite class of 4-regular expander graphs, and let Q be the class of all subdivisions of graphs in ∆, that is Q = {H2 | H ∈ ∆}, where H2 is obtained from H by subdividing each edge once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We start by establishing an easy but convenient fact on the treewidth of the graphs in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Q has unbounded treewidth and tw(H2) = Θ(|V (H)|) for H ∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' As in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1, tw(H) = Θ(|V (H)|) for H ∈ ∆, since expanders have treewidth linear in the number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since H is a minor of H2, and since taking minors cannot increase treewidth (see [14, Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='7]), we thus have that tw(H2) = Ω(|V (H)|)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Finally, we have tw(H2) ≤ |V (H2)| since the treewidth is at most the number of vertices, and |V (H2)| = O(|V (H)|) since H is 4-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In combination, we obtain tw(H2) = Θ(|V (H)|) for H ∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that this also implies that Q has unbounded treewidth (as ∆ is infinite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ◀ For what follows, given a subdivision H2 of a graph H, it will be convenient to assume that V (H2) = V (H) ∪ SE, where SE = {se | e ∈ E(H}) is the set of the subdivision vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Definition 23 (Odd Fractures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let H ∈ ∆ and let τ be a fracture of H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We say that τ is odd if the following two conditions are satisfied: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each s ∈ SE the partition τs consists of two singleton blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each v ∈ V (H) the partition τv consists of two blocks of size 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Consider Figure 3 for a depiction of an odd fracture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The following two lemmas are crucial for our construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Lemma 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let H ∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The number of odd fractures of H2 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The first condition in Definition 23 leaves only one choice for subdivision vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let us thus consider a vertex v ∈ V (H) = V (H2) \\ SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since H is 4-regular, there are 4 incident edges to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Now note that there are precisely 3 partitions of a 4-element set with two blocks of size 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus the total number of odd fractures of H2 is 3|V (H)|, which is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ◀ ▶ Lemma 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let H ∈ ∆, let k = 2|V (H)| and let τ be a fracture of H2 such that τv consists of at most 2 blocks for each v ∈ V (H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then H2 ♯ τ ∼= kP2 if and only if τ is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' First observe that |E(H2)| = 2|E(H)| = 4|V (H)| = 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus the number of edges of H2 ♯ τ is equal to 2k (for each fracture τ of H2), which is also equal to the number of edges of kP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Goldberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Roth 13 Figure 3 (Top:) Subdividing a 4-regular expander in ∆ depicted by the neighbourhood of an individual vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (Centre:) Illustrations of odd fractures (Definition 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each non-subdivision vertex, there are only three ways to satisfy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' in Definition 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This observation is used in Lemma 24 to show that the number of odd fractures is a power of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (Bottom:) Elements of ColSub(kP2 → (G, cE)) inducing fractures of H2 such that each partition has at most two blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Lemma 25 shows that those are precisely the odd fractures of H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 14 Parameterised and Fine-grained Subgraph Counting, modulo 2 Thus, H2 ♯ τ is isomorphic to kP2 if and only if each connected component of H2 ♯ τ is a path of length 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' It follows immediately by Definition 23 that τ being odd implies that H2 ♯ τ consists only of disjoint P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' It thus remains to show the other direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Assume for contradiction that there is a subdivision vertex s ∈ SE of H2 such that τs consists of only one block (recall that s has degree 2, thus τs either consists of two singleton blocks, or of one block of size 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let e = {u, v} ∈ E(H) be the edge corresponding to s, that is, s was created by subdividing e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since H2 ♯ τ is a union of P2, we can infer that τv and τu contain a singleton block (otherwise we would have created a connected component which is not isomorphic to P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Now recall that both u and v have degree 4, since H is 4-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We obtain a contradiction as follows: By assumption of the lemma, we know that τv and τu can have at most two blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since we have just shown that both contain a singleton block, it follows that both τv and τu contain one further block of size 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' However, a block of size 3 yields a vertex of degree 3 in the fractured graph H2 ♯ τ, contradicting the fact that H2 ♯ τ consists only of disjoint P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus we have established that, for each s ∈ SE, the partition τs consists of two singleton blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Given this fact, the only way for H2 ♯ τ being a disjoint union of P2 is that each partition τv, for v ∈ V (H) = V (H2) \\ SE, consists of two blocks of size 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ◀ We are now able to prove our hardness result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Lemma 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The problem ⊕Sub(FP2) is ⊕W[1]-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Furthermore, on input kP2 and G, the problem cannot be solved in time f(k) · |G|o(k/ log k) for any function f, unless rETH fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We reduce from ⊕cp-Hom(Q), which, by Lemma 22 and Theorem 12, is ⊕W[1]-hard and for H′ ∈ Q, it cannot be solved in time f(|H′|) · |G|o(|V (H′)|/ log |V (H′)|), unless rETH fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let H′ and (G, c) be an input instance to ⊕cp-Hom(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' There is an algorithm that takes as input a graph H′ ∈ Q and finds a graph H ∈ ∆ with H′ = H2 — this is basically 2-colouring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The run time of this algorithm depends on |H′| but clearly not on (G, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let k = 2|V (H)| and note that |E(H2)| = 2|E(H)| = 4|V (H)| = 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We consider the graph G as a 2k-edge-coloured graph, coloured by cE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' That is, each edge e = {x, y} of G is assigned the colour cE(e) = {c(x), c(y)} which is an edge of H′ = H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Now, for any H2-coloured graph (G′, c′) recall that ColSub(kP2 → (G′, c′ E)) is the set of subgraphs of G′ that are isomorphic to kP2 and that include each edge colour (each edge of H2) precisely once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We will see later that ⊕ColSub(kP2 → (G′, c′ E)) can be computed using our oracle for ⊕Sub(FP2) using the principle of inclusion and exclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' It was shown in [29, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1] that there is a unique function a such that, for every H2-coloured graph (G′, c′), #ColSub(kP2 → (G′, c′ E)) = � ρ a(ρ) · Hom(H2 ♯ ρ → (G′, c′)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (5) where the sum is over all fractures of H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' As in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1 from [29, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='3] we know that a(⊤) = � ρ∈F(kP2,H2) � w∈V (H2) (−1)|ρw|−1 · (|ρw| − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , (6) where ⊤ is the fracture in which each partition consists only of one block and F(kP2, H2) is the set of all fractures ρ of H2 such that H2 ♯ ρ ∼= kP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Our next goal is to show that a(⊤) = 1 mod 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' First, suppose that a fracture ρ contains a partition ρw with at least three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then (|ρw| − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' = 0 mod 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus such fractures L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Goldberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Roth 15 do not contribute to a(⊤) if arithmetic is done modulo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Next, note that if, for each w, the partition ρw contains at most 2 blocks, then � w∈V (H2) (−1)|ρw|−1 · (|ρw| − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' = 1 mod 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let Odd(kP2, H2) be the set of all fractures ρ of H2 such that H2 ♯ ρ ∼= kP2 and each partition of ρ consists of at most 2 blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Our analysis then yields a(⊤) = |Odd(kP2, H2)| mod 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Finally, Lemma 25 states that Odd(kP2, H2) is precisely the set of odd fractures, and Lemma 24 thus implies that |Odd(kP2, H2)| = 1 mod 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Consequently, a(⊤) = 1 mod 2 as well, and we have achieved the goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Next we can proceed similarly to the case of triangle packings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' As in that case, the goal is to compute ⊕Hom((H2, idH2) → (G, c))) which is equal to ⊕Hom((H2 ♯ ⊤, c⊤) → (G, c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since a(⊤) is odd, we can invoke Lemma 14 to recover this term by evaluating the entire linear combination (5), that is, if we can evaluate the function ⊕ColSub(kP2 → ⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This can be done by using Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Each call to the oracle is of the form ⊕Sub(kP2 → ˆG) where | ˆG| is bounded by f(k) · |G|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Now recall that k ∈ Θ(|V (H)|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' By Lemma 22, we thus have k = Θ(tw(H2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Hence our reduction yields ⊕W[1]-hardness and transfers the conditional lower bound under rETH as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ◀ We can now conclude the treatment of hereditary pattern classes by proving Theorem 4, which we restate for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let H be a hereditary class of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If H is matching splittable, then ⊕Sub(H) is fixed-parameter tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Otherwise, the problem is ⊕W[1]-complete and, assuming rETH, cannot be solved in time f(|H|) · |G|o(|V (H)|/ log |V (H)|) for any function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The fixed-parameter tractability result was shown in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For the hardness result, using the fact that H is not matching splittable and Theorem 16 we obtain four cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If H contains all cliques or all bicliques, then hardness follows from Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If H contains all triangle packings, then hardness follows from Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If H contains all P2-packings, then hardness follows from Lemma 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since the case distinction is exhaustive, the proof is concluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ◀ 4 Classification for Trees Our overall goal is to prove Theorem 5, which we restate for convenience: ▶ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let T be a recursively enumerable class of trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If T is matching splittable, then ⊕Sub(T ) is fixed-parameter tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Otherwise ⊕Sub(T ) is ⊕W[1]-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We start by introducing some terminology for trees which will be used in the remainder of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Definition 27 (2-paths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A 2-path of length a of a tree T is a path x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , xa such that deg(x0) ̸= 2, deg(x1) = · · · = deg(xa−1) = 2 and deg(xa) ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Next we introduce rays, which are restricted 2-paths that will be crucial in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 16 Parameterised and Fine-grained Subgraph Counting, modulo 2 ▶ Definition 28 (source, ray, degL,a, degL, and degNL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let T be a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A source of T is any vertex with degree greater than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A ray of length a of T is a 2-path x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , xa such that deg(x0) > 2 and deg(xa) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We call x0 the source of the ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Given a vertex s of degree at least 3, we write degL,a(s) for the number of rays of length a with source s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We set degL(s) := � a degL,a(s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Finally, we set degNL(s) := deg(s) − degL(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Next, we introduce parameters Fa,b, Sc and Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Our goal is then to show that, for every non-matching-splittable class of trees, at least one of those two parameters is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Definition 29 (Forks and Fa,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let a, b be positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A source s of a tree T is called an a-b-fork if degNL(s) = 1 and one of the following is true a ̸= b and degL,a(s), degL,b(s) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' a = b and degL,a(s) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The a-b-fork number of T, denoted by Fa,b(T) is the maximum size of an independent set containing only a-b-forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Finally, we say that a class of trees T has unbounded fork number if for every positive integer B there are positive integers a and b and a tree T ∈ T such that Fa,b(T) ≥ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Definition 30 (Stars and Sc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A star of size k > 1 in a tree T is a collection of k distinct rays that have a common source s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For a positive integer c ≥ 3, a c-star of size k in a tree T is a collection of k distinct rays of length c that have a common source s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The c-star number of a tree T, denoted by Sc(T) is the maximum size of a c-star in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Finally, we say that a class of trees T has unbounded star number if for every positive integer B there exists c ≥ 3, and a tree T ∈ T such that Sc(T) ≥ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Definition 31 (C-gadgets and Cd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A C-gadget4 of order d and length k in a tree T is a path x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , xk such that one of the following is true for each inner vertex xi ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , k −1}: (i) deg(xi) = 2, that is N(xi) = {xi−1, xi+1}, or (ii) xi is a source and every neighbour v ∈ N(xi)\\{xi−1, xi+1} is contained in a ray of length at most d from xi to a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The Cd-number of a tree T, denoted by Cd(T) is the length of the longest C-gadget of order d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Finally, we say that a class of trees T has unbounded C-number if there exists d > 0 such that for every positive integer B, and a tree T ∈ T such that Cd(T) ≥ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that the ordering of the quantifiers in the definition of the Cd-number is different from the ordering in the definition of the c-star-number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This is due to technical reasons which are important for the proof of Lemma 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Lemma 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let T be a class of trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If T is not matching splittable, then T has either unbounded fork number, unbounded star number, or unbounded C-number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We can assume that there is an overall bound d on the length of 2-paths in trees in T : Otherwise, T already has unbounded C-number (see (i) in Definition 31)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Hence the length of every ray in any tree in T is bounded by d as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus T has unbounded fork number if and only if for every positive integer B there are a, b ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , d} and a tree T ∈ T such that Fa,b(T) ≥ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 4 C stands for caterpillar, the shape of which resembles the structure of a C-gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Goldberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Roth 17 T has unbounded C-number if and only if Cd is unbounded in T (see Definition 31)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' T has unbounded star number if and only if for every positive integer s there is a c ∈ {3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , d} and a tree T ∈ T such that Sc(T) ≥ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We split the proof into two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' T has unbounded diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In Case 1, we show that T has unbounded fork number or unbounded C-number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If Cd is unbounded in T then T has unbounded C-number and we are done so assume that there is a constant h such that Cd(T) ≤ h for every T ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Now let B be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We show that there are a, b ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , d} and T ∈ T with Fa,b(T) ≥ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' To this end, we use the premise that T has unbounded diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let k > (h + 2)(Bd2 + 1) be a positive integer, and let T ∈ T be such that there is a path P = s, p0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , pk, t in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Observe that the deletion of all edges in P decomposes T into a family of disjoint subtrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We write Ti for the subtree that contains pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Now decompose P into segments P1, P2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' of length h + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that a segment Pj = pj0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , pjh+2 yields a C-gadget of order d and length > h if and only if Tji is either a star or an isolated vertex for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , h + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since no such C-gadgets exist by assumption, we obtain that each segment Pj of the path P contains a vertex pij such that Tij is neither a star nor an isolated vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Assume that Tij is rooted at pij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since Tij is neither an isolated vertex nor a star, there must be a (proper) descendant vij of pij (in Tij) such that vij is an (aij, bij)-fork for some aij, bij ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Now note that there are at most d2 pairs of integers in {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since we have at least one fork for every segment and since there are at least ⌊k/(h + 2)⌋ > Bd2 + 1 segments, we thus obtain by the pigeon-hole principle that there is a pair a, b ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , d} such that, for at least B segments Pij, the node vij is an (a, b)-fork in Tij and thus also in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since those forks are pairwise non-adjacent, we obtain, as desired, that the (a, b)-fork number of T is at least B, concluding Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' T has bounded diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let D be the assumed upper bound on the diameter of trees in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If T has unbounded star number then we are finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Assume instead that T has bounded star number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then there is a positive integer s such that for all c ∈ {3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , d} and every tree T ∈ T , Sc(T) < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We will show that T has unbounded fork number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Consider any positive integer B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We will show that there are a, b ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , d} and T ∈ T with Fa,b(T) ≥ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let k > (D+1)(Bd2 +1)(d2s+1) be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since T is not matching splittable, there is a tree T ∈ T whose matching-split number is at least k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that T is not a path since every path with matching-split number at least k has length greater than k > D, contradicting the bound on the diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Now fix any vertex r of T as the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Given a vertex v of T, we write Tv for the subtree rooted at v (assuming that r is the overall root).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We call v a rooted fork if Tv is a star — observe that each rooted fork must indeed be a fork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let f be the number of rooted forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Similar to the argument in Case 1, if f > Bd2 + 1, then by the pigeon-hole principle there are a, b ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , d} such that Fa,b(T) ≥ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Hence assume for contradiction that f ≤ Bd2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let R be the set of all rays of T and recall that each ray in R is, by definition, a 2-path of the form R = x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , xd′ for d′ ≤ d, where deg(x0) > 2 and xd′ is a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We call a ray R long if d′ ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that the source of every ray must either be a rooted fork, or it must lie on a path from the root r to one of the rooted forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let T ′ be the subtree of T induced by all vertices that lie on paths between r and a rooted fork (including r and all rooted forks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since there are f rooted forks and the depth 18 Parameterised and Fine-grained Subgraph Counting, modulo 2 of T is bounded by D, |V (T ′)| ≤ (D + 1)f ≤ (D + 1)(Bd2 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Consider a vertex v of T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Assume for contradiction that v is the source of > ds long rays (in T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Recall that for all c ∈ {3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , d} we have that Sc(T) < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Recall further that each long ray has length d′ for some 3 ≤ d′ ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus we obtain a contradiction by the pigeon-hole principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Now let S be the set containing all vertices of T ′ and all vertices of long rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Noting that each long ray has length at most d, and that the source of each long ray must be a vertex of T ′ by construction, we can use the observation that each vertex of T ′ is the source of at most ds long rays to (generously) bound |S| ≤ |V (T ′)| + |V (T ′)| · d · ds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note further that T[V (T) \\ S] consists only of isolated edges and vertices: The only vertices in V (T) \\ S are non-source vertices of rays of length < 3, the sources of which are in T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus, S is a splitting set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Finally, recalling that |V (T ′)| ≤ (D + 1)f ≤ (D + 1)(Bd2 + 1), we have |S| ≤ |V (T ′)| + |V (T ′)| · d · ds ≤ (D + 1)(Bd2 + 1)(d2s + 1) , contradicting the fact that the matching-split number of T is strictly larger than (D + 1)(Bd2 + 1)(d2s + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This concludes Case 2, and hence the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ◀ In the next three subsections, we will prove hardness of ⊕Sub(T ) for non-matching- splittable T in each of the three cases given by Lemma 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1 Unbounded C-number For our hardness proof, it will be useful to find a proper sub-gadget of a C-gadget in a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Definition 33 (Strong C-gadgets, junctions, and closedness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let C = x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , xL be a C-gadget of order d and length L in a tree T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We call C a strong C-gadget with k junctions if there are integers 0 = i0 < i1 < · · · < ik < ik+1 = L such that (I) for all j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , k}, ij+1 − ij > 2d, and (II) for all j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , k}, xij is the source of a ray Rj of length d that does not contain one of the neighbours xij−1 and xij+1 of xij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The vertices xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , xik are called the junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Finally, a strong C-gadget is called closed if neither xi1 nor xik are forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='5 Consider the bottom part of Figure 4 for a visualisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We start with the following lemma which establishes the existence of a strong C-gadget with many junctions inside a long enough C-gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Lemma 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let T be a tree such that the longest 2-path in T has length d ≥ 1, and let k be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then there exists L > 0 (only depending on k and d) such that the following is true: If T contains an C-gadget of order d and length L, then there exists 1 ≤ d′ ≤ d such that T contains a strong C-gadget of order d′ with at least k junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let f(x) = x/(k + 1) − 2d − 1 and let L be large enough such that f d(L) > d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let Hd = x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , xL be a C-gadget of order d and length L in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let d′ = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that Hd′ is a C-gadget of order d′ and length at least L = f d−d′(L) in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each graph Hd′ with d′ ≥ 1 we will either 5 The condition of being closed rules out the special case in which x0 or xL are leaves of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' More generally it rules out the case where there is a ray from x1 including x0 or from xk including xL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Goldberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Roth 19 (1) construct a strong C-gadget with k junctions with order d′, or (2) find a subsequence Hd′−1 of Hd′ that is an C-gadget of order d′ − 1 of length at least f d−(d′−1)(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If we ever do (1) we are finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If from d′ = 1 we do (2) then we find a 2-path of length at least f d(L) > d, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Here is how we proceed from Hd′ = y0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , yℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We set i0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then iteratively, for each j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , k} we will either construct Hd′−1 as in (2) or we find ij ∈ {ij−1 + 2d + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , ℓ} such that yij is the source of a length-d′ ray that does not contain yij − 1 or yij + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If we succeed in defining i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , ik, ik+1 in this way then y0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , yik+1 is a strong C-gadget with k junctions of order d′ so (1) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let us now make this argument rigorous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' again, assume that Hd′ = y0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , yℓ is a C- gadget of order d′ and length ℓ ≥ f d−d′(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Set i0 = 0 and, starting with j = 0, proceed iteratively as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let Sj be the set of all indices i ∈ {ij−1 + 2d + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , ℓ} such that yi is the source of a length-d′ ray that does not contain yi−1 and yi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If Sj = ∅ then set stop = j and terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Otherwise, set ij = min Sj and j ← j + 1, and go back to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We now distinguish two cases: If stop ≥ k + 1, then we found indices i0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , ik+1 such that ˆHd′ := y0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , yik+1 is a strong hardness gadget of order d′ with k junctions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' hence we achieved (1) and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Otherwise we have stop < k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let Ij := {ij, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , ij+1 − 1} for all 0 ≤ j < stop, and let Istop = {istop, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , ℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' By the pigeon-hole principle, at least one of those intervals, say Ij′, has size at least ℓ/(stop + 1) ≥ ℓ/(k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Now, by construction of our iterative procedure above, we find that the sub-interval {ij′ + 2d + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , ij′+1 − 1} ⊆ Ij′ contains no index i such that yi is the source of a length-d′ ray that does not contain yi−1 and yi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus, the subsequence Hd′−1 := yij′+2d+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , yij′+1−1 constitutes a C-gadget of order d′ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Furthermore, Hd′−1 has length at least ℓ/(k + 1) − 2d − 1 = f(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since ℓ ≥ f d−d′(L), and since f is monotonically increasing, we find that f(ℓ) ≥ f d−(d′−1)(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Hence we achieved (2) and we can conclude this case as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ◀ Now, by removing the first and the last junction, we can also ensure the existence of a closed strong C-gadget ▶ Corollary 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let T be a tree such that the longest 2-path in T has length d ≥ 1, and let k be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then there exists L > 0 (only depending on k and d) such that the following is true: If T contains an C-gadget of order d and length L, then there exists 1 ≤ d′ ≤ d such that T contains a closed strong C-gadget of order d′ with at least k junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Use Lemma 34 with k + 2 rather than k and observe that every strong C-gadget with k + 2 junctions also yields a closed strong C-gadget with k junctions by removing i1 and ik+2 from the list of indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since xi1 and xik+2 must have degree at least 3 (they are inner vertices of a C-gadget and they are junctions), we obtain that neither xi2 and xik+1 can be forks of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ◀ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1 Constructions of Q and ˆG For the scope of this subsection, to avoid notational clutter, we assume the following are given: Positive integers k and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 20 Parameterised and Fine-grained Subgraph Counting, modulo 2 A tree T that contains a closed strong C-gadget H = x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , xℓ of order d with k junctions xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , xik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Additionally, for each j ∈ [k], we fix a ray Rj = xij, r1 j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , rd j of length d, the source of which is xij and which does not contain one of the neighbours xij−1 and xij+1 — note that the Rj must exist as the xij are junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A k-vertex cubic graph ∆ containing a Hamiltonian cycle v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , vk, v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We emphasise that the set of edges of ∆ not contained in the Hamilton cycle must constitute a perfect matching, that is, a set of k/2 pairwise non-incident edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This must be satisfied since ∆ is cubic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Definition 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The core of H, denoted by C(H), contains the subsequence xi1, xi1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , xik−1, xik and the vertices of the rays Rj, that is C(H) := {xi1, xi1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , xik−1, xik} ∪ k� j=1 V (Rj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Definition 37 (Q(∆, T, H) and τQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Set ℓj := ij+1 − ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The graph Q = Q(∆, T, H) is obtained from ∆ as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The edge {vk, v1} is deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , k − 1} the edge {vj, vj+1} is replaced by a path of length ℓj: Pj = vj, u1 j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , uℓj−1 j , vj+1 , where the ut j are fresh vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Each edge e = {vi, vj} not contained on the Hamilton cycle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=', j /∈ {i − 1, i + 1}, is replaced by a path Pi,j of length 2d: Pi,j = vi, w1 i , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , wd−1 i , m(e), wd−1 j , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , w1 j, vj , where the wt i and wt j are fresh vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Finally τ = τ(∆, T, H) is a fracture of Q defined as follows: For each m(e), the partition τm(e) contains two singleton blocks, and for all remaining vertices v of Q the partition τv only contains one block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since ∆, T and H are fixed in this subsection, to avoid notational clutter, we just write Q and τ, rather than Q(∆, T, H) and τ(∆, T, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' It turns out that Q is isomorphic to a quotient graph of T[C(H)] obtained by identifying the endpoints of the rays Ri and Rj for every {vi, vj} ∈ E(∆) with j /∈ {i − 1, i + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This induces a homomorphism from T[C(H)] to Q that will be useful in the construction of ˆG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' hence we explicitly define this mapping below: ▶ Definition 38 (γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We define a function γ : C(H) → V (Q) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We map the sequence xi1, xi1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , xik−1, xik in C(H) to the sequence v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , vk in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' More precisely, for each j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , k − 1} and t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , ℓj − 1}, we set γ(xij) := vj and γ(xij+t) := ut j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each edge e = {vi, vj} of ∆ with j /∈ {i − 1, i + 1}, we map V (Ri) and V (Rj) to the path Pi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' More precisely, for each t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , d − 1} we set γ(rt i) := wt i and γ(rt j) = wt j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Furthermore, we set γ(rd i ) := m(e) =: γ(rd j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (Note that the images of the sources of the rays Ri and Rj are already set in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=') ▶ Observation 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The function γ is an edge-bijective homomorphism from T[C(H)] to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let us provide the induced egde-bijection explicitly: L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Goldberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Roth 21 ▶ Definition 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (E′, γE) Define E′ := E(T[C(H)]), that is, E′ ⊆ E(T) contains all edges on the sub-path xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , xik of H and all edges of the rays R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We write γE : E′ → E(Q) for the edge-bijection from E′ to E(Q) induced by the homomorphism γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Now let (G, c) be a Q-coloured graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We state the following fact explicitly, since it will be crucial in our construction: ▶ Observation 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let (G, c) be a Q-coloured graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The mapping cE ◦ γ−1 E is a map from E(G) to E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Our goal is to construct a graph ˆG = ˆG(G, c, T, H) from G, and an edge-colouring ˆγ : E( ˆG) �→ E(T) whose range is E(T) such that ⊕Emb((Q ♯ τ, cτ) → (G, c)) = ⊕ColSub(T → ( ˆG, ˆγ)), that is, the number of colour-preserving embeddings from the fractured graph Q ♯ τ to (G, c) is equal, modulo 2, to the number of subgraphs of ˆG that are isomorphic to T and that contain each edge-colour in E(T) precisely once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For what follows, let V (R) := ∪k j=1V (Rj) be the set of all vertices of the rays R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We are now able to define ˆG = ˆG(G, c, T, H);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' the construction is illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The definition uses the function cE introduced in Definition 6 and the functions γ and γE introduced in Definitions 38 and 40, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' It also uses the mapping cE ◦ γ−1 E from E(G) to E′ (see Observation 41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Definition 42 ( ˆG(G, c, T, H), ˆγ(G, c, T, H)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let (G, c) be a Q-coloured graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The pair ( ˆG, ˆγ) = ( ˆG(G, c, T, H), ˆγ(G, c, T, H)) is an edge-coloured graph constructed as follows, where the co-domain of ˆγ is E(T): (A) The graph ˆG contains G as a subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each e ∈ E(G), define ˆγ(e) = γ−1 E (cE(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (B) The vertex set of ˆG is the union of V (G) and V (T) \\ C(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (C) Pairs of vertices in V (T) \\ C(H) are connected by an edge in ˆG if and only if they are adjacent in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each such edge e, ˆγ(e) = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (D) The remaining edges of ˆG are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each edge e ∈ E(T) that connects a vertex z ∈ V (T) \\ C(H) to a vertex y ∈ C(H) there are corresponding edges in ˆG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' These edges connect z to all vertices g ∈ V (G) such that c(g) = γ(y) For each such edge e′ in ˆG, ˆγ(e′) = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Observe that for each element Tcol ∈ ColSub(T → ( ˆG, ˆγ)) the induced subgraph Tcol[G] := Tcol[V (Tcol) ∩ V (G)] of Tcol is an edge-colourful subgraph in G, that is, Tcol[G] contains precisely one edge per edge-colour of G under the edge colouring ˆγ hence it contains precisely one edge per edge- colour of G under cE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' As shown in Section 3 in the full version [30] of [31], Tcol[G] thus induces a fracture ρ = ρ(Tcol) of Q: Two edges {v, w} and {v, y} of Q are in the same block in the partition ρv corresponding to vertex v of Q if and only if the edges of Tcol[G] that are coloured γ−1 E ({v, w}) and γ−1 E ({v, y}) are adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In what follows, we show that ρ must always be equal to τ(∆, T, H) (see Definition 37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Lemma 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For every Tcol ∈ ColSub(T → ( ˆG, ˆγ)) we have that ρ(Tcol) = τ(∆, T, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' To avoid notational clutter, we set ρ := ρ(Tcol) and τ := τ(∆, T, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let T1 and T2 be the subtrees of T attached to the ends of the C-gadget H as shown in the bottom part of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 22 Parameterised and Fine-grained Subgraph Counting, modulo 2 Figure 4 (Below): The tree T containing a closed strong C-gadget of order d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' the green dashed lines are rays of length d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (Left): The construction of ˆG = ˆG(G, c, T, H);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' note that the removal of the vertices and edges coloured blue yields G (see Definition 42), and note that G is Q-coloured as depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (Right): The graphs ∆ and Q = Q(∆, T, H);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' we assume in the picture that {v2, vk−1} is an edge of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We first give an overall intuition of the proof;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' consider Figure 5 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since Tcol is isomorphic to T, there must be a (unique) path connecting T1 and T2 in ˆG (recall L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Goldberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Roth 23 that, since Tcol is edge-colourful and since every edge in T1 and T2 has a different colour — see (C) in Definition 42 — Tcol must contain all edges in T1 and T2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We claim that this path must follow the outer cycle in ˆG, in which case the designated rays in R of length d at the junctions must follow the inwards direction and thus induce τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' To see why the path connecting T1 and T2 must follow the outer cycle, first recall that Vj is the subset of V (G) coloured by c with vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then recall that the path between Vj and Vj+1 along the outer cycle in ˆG has length ℓj ≥ 2d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Hence the designated rays in R cannot be used to cover all edge colours in the path between Vj and Vj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We next provide a rigorous argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let S := V (T1) ∪ V (T2) ∪ {x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , xi1−1} ∪ {xik+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , xk+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that S is a subset of V (T) \\ V (H) hence it is a subset of V (T) and of V ( ˆG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We first claim that every fork and every ray of length > d of T must be fully contained in the subgraph of T induced by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This claim follows from the definition of closed strong C-gadgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In particular, the condition of being closed implies that neither xi1 nor xik is a fork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' As a consequence, every fork and every ray of length greater than d of Tcol must be contained in the subgraph of ˆG induced by S as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Additionally, this implies that none of the vertices in Tcol[G] can be a fork or the source of a ray of length > d in Tcol — otherwise, Tcol would have either more forks or more rays of length > d than T, contradicting the fact that Tcol and T are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Recall that V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , Vk denote the subsets of vertices of G that are coloured by c with v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Now let P be the (unique) path P in Tcol that connects T1 with T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then, starting with V1 and ending with Vk, the path P must pass through a sequence of colour classes V1 = Vj1, Vj2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , Vjt = Vk of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The following claim formalises the idea that this sequence must correspond to the Hamilton cycle v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , vk in ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Claim: We have t = k and Vji = Vi for each i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Before proving the claim, we show that it implies the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since, from the claim, P must follow the outer cycle, the fracture ρ = ρ(Tcol) induced by Tcol must split the inner paths of length 2d (otherwise Tcol would contain a cycle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' However, since there are no sources or rays of length greater than d outside of S in Tcol, ρ must split all of the inner length-2d paths at the central vertex m(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Furthermore, it cannot split additional vertices since this would disconnect Tcol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus, ρ is the fracture τ, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ■ To conclude the proof, we now prove the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note first that P cannot pass through any of the colour classes Vi more than once as this would cause Tcol to use an edge-colour multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Next assume for contradiction that P misses some colour class Va for some a ∈ [2, k − 1] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=', we assume that t < k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since Tcol is a connected tree containing all of the edge colours in Q there must be an index ji ̸= a and a vertex u ∈ Vji ∩ P such that Tcol contains a (unique) path Pu from u to a vertex w ∈ Va.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In order to get the contradiction, root Tcol at u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Construct a subtree Tcol(u) of Tcol as follows: For each neighbour x of u except the ancestor of w on the path from u, we delete x and all of its descendants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Observe that the edge colours of Tcol(u) are disjoint from the edge-colours of P and that V (Tcol(u)) is disjoint from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Now, if Tcol(u) is a path, then (using that ℓi > 2d), we obtain that u is the source of a ray in Tcol of length greater than d, contradicting the fact that every ray of length > d of Tcol is in the subgraph of ˆG induced by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Otherwise, Tcol(u) contains a fork, contradicting the fact that all forks of Tcol are in the subgraph of ˆG induced by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Having established that t = k and that no Vi is visited more than once, it remains to show that P visits the colour classes in the correct order, that is Vji = Vi for each i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 24 Parameterised and Fine-grained Subgraph Counting, modulo 2 Figure 5 Illustration of Lemma 43: The only possibility for an edge-colourful copy of T to be embedded in ˆG is depicted in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Assume for contradiction that this is not the case, which allows us to set m := min{i ∈ [k] | Vji ̸= Vi} − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that m ≥ 1 since j1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let zm ∈ Vm ∩ P and zm+1 ∈ Vm+1 ∩ P and recall that G contains colour classes U 1 m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , U ℓm−1 m corresponding to the path Pm = vm, u1 m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , uℓm−1 m , vm+1 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Goldberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Roth 25 in Q (see Definition 37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let us now define the subtrees Tcol(m) and Tcol(m + 1): For Tcol(m) we root Tcol at zm and for each neighbour x of zm in Tcol, we delete x and all of its descendants unless x ∈ U 1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For Tcol(m + 1) we root Tcol at zm+1 and for each neighbour x of zm+1 in Tcol, we delete x and all of its descendants unless x ∈ U ℓm−1 m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that at least one of Tcol(m) and Tcol(m + 1) must have depth greater than d (if rooted at zm and zm+1, respectively), since ℓm > 2d and Tcol is edge-colourful with respect to ˆγ, that is, we have to make sure that we cover all of the edge colours {vm, u1 m}, {u1 m, u2 m}, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , {uℓm−1 m , vm+1} Finally, regardless of which one of the two subtrees has depth greater than d, we will find either a fork, or the source of a ray of length greater than d outside of the set S, yielding the desired contradiction and concluding the proof of the claim, and hence the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ◀ We are now able to prove the main lemma of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Lemma 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ⊕Emb((Q ♯ τ, cτ) → (G, c)) = ⊕ColSub(T → ( ˆG, ˆγ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We start with the following claim from [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Claim: A colour-preserving embedding ϕ ∈ Emb((Q ♯ τ, cτ) → (G, c)) is uniquely defined by its image (which is a subgraph of (G, c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For convenience, we provide a proof of the claim: Consider in image (G′, c′) of ϕ where G′ is a subgraph of G and c′ = c |V (G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let e = {u, v} be an edge of G′ Then c′(e) = {c(u), c(v)} is an edge of Q since c is a Q-colouring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Recall that Q ♯ τ is Q-coloured by the function cτ that maps wB to w for each w ∈ V (Q) and block B ∈ τw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Now recall the definition of fractured graphs (Definition 8) and let B1 and B2 be the blocks of τc(u) and τc(v) that contain c(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then, since ϕ is an embedding, it maps c(u)B1 to u and c(v)B2 to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since Q does not have isolated vertices, continuing this process over all edges of G′ defines ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This concludes the proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ■ By the claim, it is sufficient to construct a bijection b from elements in ColSub(T → ( ˆG, ˆγ)) to subgraphs (G′, c′) that are images of embeddings in Emb((Q ♯ τ, cτ) → (G, c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Given Tcol ∈ ColSub(T → ( ˆG, ˆγ)) we set b(Tcol) := (Tcol[G], c(Tcol)) where c(Tcol) is the colouring of vertices of Tcol[G] which agrees with ˆγ on the edges of Tcol[G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In the rest of the proof, we show that b is the desired bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' First, we have to show that for all Tcol, (Tcol[G], c(Tcol)) is the image of an embedding in Emb((Q ♯ τ, cτ) → (G, c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' To this end, recall that Tcol[G] induces a fracture ρ = ρ(Tcol) of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' By the definition of ρ, Tcol[G] and Q ♯ ρ are isomorphic and this isomorphism preserves the colours so cρ agrees with ˆγ on the edges of Q ♯ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This implies that cρ and c(Tcol) are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' So (Tcol[G], c(Tcol)) is the image of an embedding in Emb((Q ♯ ρ, cρ) → (G, c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Finally, Lemma 43 guarantees that ρ = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Second, we will show that b is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' To this end, let Tcol1 ̸= Tcol2 ∈ ColSub(T → ( ˆG, ˆγ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since Tcol1 and Tcol2 must both fully contain V (T) \\ C(H), and since both are edge-colourful (see Definition 42), the only possibility for Tcol1 and Tcol2 not being equal is that they disagree on G, that is, Tcol1[G] ̸= Tcol2[G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This proves b to be injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Finally, we will show that b is surjective: Given any (G′, c′) that is the image of an embedding ϕ ∈ Emb((Q ♯ τ, cτ) → (G, c)), we construct Tcol(G′, c′) ∈ ColSub(T → ( ˆG, ˆγ)) with b(Tcol(G′, c′)) = (G′, c′) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Observe first that G′ is isomorphic to T[C(H)] since 26 Parameterised and Fine-grained Subgraph Counting, modulo 2 Q ♯ τ is, by definition of τ, isomorphic to T[C(H)]: Splitting the inner paths of length 2d in Q at their central vertices yields precisely T[C(H)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then Tcol(G′, c′) is obtained by adding the remainder of T to (G′, c′): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We add to (G′, c′) all vertices in V (T) \\ C(H) (see (B) in Definition 42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We add all edges between vertices in V (T) \\ C(H) that are present in ˆG (see (C) in Definition 42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Finally, we connect a vertex in z in V (T) \\ C(H) with a vertex w in G′ if and only if z and w are connected in ˆG (see (D) in Definition 42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The resulting subgraph Tcol(G′, c′) of ˆG is clearly edge-colourful and isomorphic to T, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ◀ We are now able to establish hardness of ⊕Sub(T ) in case of unbounded C-number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Lemma 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let T be a recursively enumerable class of trees of unbounded C-number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then ⊕Sub(T ) is ⊕W[1]-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Assume first that T contains trees with 2-paths of unbounded length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In this case we reduce from the problem of counting k-cycles, modulo 2, which was shown ⊕W[1]-hard in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In the first step, this problem reduces to the problem of counting s-t-paths of length k, modulo 2 as shown in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='2 in the full version [28] of [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In the second and final step, we can easily reduce from the problem of counting s-t-paths of length k, modulo 2, to ⊕Sub(T ), as shown in Figure 6: Concretely, let (G, s, t, k) be a problem instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since T contains trees with 2-paths of unbounded length, we can find, in time only depending on k, a tree T in T containing a 2-path x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , xk+1, xk+2 of length k + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let furthermore T1 and T2 be the subtrees of T as depicted in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We construct a graph G′ from G in two steps as follows: First, we add fresh vertices x0 and xk+2 and edges {x0, s} and {t, xk+2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Second, we add T1 and T2 and identify their roots with x0 and xk+2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The construction is depicted in Figure 6 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Now let A be the set of subgraphs of G′ that are isomorphic to T and that contain all edges of T1 and T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' It is easy to see that the cardinality of A is equal to the number of s-t-paths of length k in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus it suffices to compute |A| mod 2, using an oracle for ⊕Sub(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This can be achieved by a simple application of the inclusion-exclusion principle: Setting S = E(T1) ∪ E(T2), we have |A| = � J⊆S (−1)|J| · #Sub(T → G′ \\ J) , (7) where G′ \\ J is the graph obtained from G′ by deleting all edges in J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We can conclude the reduction by observing that the number of terms in (7) only depends on T and thus on k, and that our oracle to ⊕Sub(T ) allows us to evaluate (7) modulo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For the remainder of the proof we can thus assume that the length of any 2-path in any tree in T is bounded by a constant d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since T has unbounded C-number, we obtain that the trees in T contain C-gadgets of order d of unbounded length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' By Corollary 35 we obtain that for any positive integer k, there is a value d′ in the range 1 ≤ d′ ≤ d such that there is a tree Tk in T which contains a strong C-gadget of order d′ with k junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let C be a class of cubic Hamiltonian graphs of unbounded treewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' that, for each k, the class C contains at most one graph with k vertices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' otherwise we just keep one k-vertex graph with the largest treewidth among all k-vertex graphs in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each ∆ ∈ C set T∆ := T|V (∆)|, that is T∆ is contained in T and contains a strong C-gadget H∆ with at least |V (∆)| junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Recall Definition 37 and set Q := {Q(∆, T∆, H∆) | ∆ ∈ C} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Goldberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Roth 27 Observe that Q(∆, T∆, H∆) contains as minor the graph obtained from ∆ by removing one edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since the removal of a single edge can decrease the treewidth only by a constant, and since treewidth is minor-monotone, we have that Q has unbounded treewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' By Theorem 12 the problem ⊕cp-Hom(Q) is therefore ⊕W[1]-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus it suffices to show that ⊕cp-Hom(Q) ≤fpt T ⊕Sub(T ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In the first step, we reduce the computation of ⊕Hom((Q, idQ) → ⋆) to the computation of ⊕Emb((Q ♯ τ, cτ) → ⋆);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' here, τ is the fracture defined in Definition 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' To this end, it was shown in [29] that ⊕Emb((Q ♯ τ, cτ) → ⋆) = � ρ≥τ µ(τ, ρ) · ⊕Hom((Q ♯ ρ, cρ) → ⋆) , (8) where the relation “≥” and the Möbius function µ are over the lattice of fractures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We omit introducing these objects in detail, since we only require that the coefficient of the term ⊕Hom((Q ♯ ⊤, c⊤) → ⋆) (which is equal to ⊕Hom((Q, idQ) → ⋆)) in the above linear combination was shown in [29] to be equal to � v∈V (Q) (−1)|τv|−1 · (|τv| − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since each partition τv has at most two blocks, the above term is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus, by Lemma 14, we can evaluate the term ⊕Hom((Q ♯ ⊤, c⊤) → ⋆) if we can evaluate the entire linear combination, that is, if we can evaluate ⊕Emb((Q ♯ τ, cτ) → ⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' It thus remains to show how we can evaluate ⊕Emb((Q ♯ τ, cτ) → ⋆) using our oracle for ⊕Sub(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' To this end, we use Lemma 44: Given any Q = Q(∆, T∆, H∆)-coloured graph (G, c) for which we want to compute ⊕Emb((Q ♯ τ, cτ) → (G, c)), we first construct ( ˆG, ˆγ) as in Definition 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then Lemma 44 yields that ⊕Emb((Q ♯ τ, cτ) → (G, c)) = ⊕ColSub(T∆ → ( ˆG, ˆγ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Finally, by Lemma 15 we can compute ⊕ColSub(T∆ → ( ˆG, ˆγ)) in FPT time using an oracle for ⊕Sub(T∆ → ⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since the size of T∆ only depends on Q, and since, with input Q we can find T∆ (recall that T is recursively enumerable) this yields indeed a parameterised Turing-reduction and the proof is concluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ◀ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='2 Unbounded Star Number We will use the same strategy as in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1: Given a tree T with large star number, we start with a properly chosen cubic graph ∆, and we construct a graph Q depending on ∆ and T which contains ∆ as a minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then we show that for any Q-coloured graph (G, c), we can construct an edge-coloured graph ( ˆG, ˆγ) such that ⊕ColSub(T → ( ˆG, ˆγ)) is equal to ⊕Emb((Q ♯ τ, cτ) → (G, c)) for a particular fracture τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' To this end, let T be a tree with star number (at least) 6k for some positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' By definition of the star number, there is a d ≥ 3 such that T contains a vertex s which is the source of 6k rays R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , R6k of length precisely d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each i ∈ [6k], let Ri = s, r1 i , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , rd i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Furthermore, let Ts be the subtree of T obtained by deleting the vertices r1 i , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , rd i for each i ∈ [6k];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' consider Figure 7 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 28 Parameterised and Fine-grained Subgraph Counting, modulo 2 Figure 6 Reduction from counting s-t-paths of length k, modulo 2, in a graph G to counting copies of a tree T with a 2-path of length at least k + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Figure 7 A tree with Sd(T) ≥ 6k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Definition 46 (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let ∆ be cubic graph on k vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We obtain Q from ∆ by substituting each vertex v by a gadget depicted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Afterwards, we connect the gadgets as follows: If {v, x} is an edge of ∆, then we identify the vertex vx in the gadget of v and the vertex xv in the gadget of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Observation 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ∆ is a minor of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The fracture τ of Q that we will be interested in is defined as follows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Figure 9 depicts the fractured graph Q ♯ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Definition 48 (τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let Q be the graph defined in Definition 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each edge {v, x} of ∆, the graph Q contains a vertex vx(= xv), which has degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We let τvx be the partition consisting of 2 singleton blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each vertex v of ∆, the vertices v3 and v5 have degree 2 in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We let τv3 and τv5 be the partitions consisting of 2 singleton blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each vertex v of ∆, the vertices v2, v4 and v6 have degree 3 in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each i ∈ {2, 4, 5} we let τvi be the partition consisting of one block of size 2 corresponding to the edges incident to vi from the left and the right, and one block of size 1 corresponding to the edge incident to vi from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Goldberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Roth 29 Figure 8 The construction of Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' the vertices v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , v6 on the gadget of v are emphasized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Figure 9 Illustration of the fractured graph Q ♯ τ via fracturing the vertex gadgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For all other vertices u of Q, we let τu be the partition consisting only of one block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Analogously to the notion of a core in the case of unbounded C-number, we will identify a specific subgraph of the tree T and we will use it to define the graph ˆG later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Definition 49 (V ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let V ′ be the vertex subset of T defined as follows: V ′ := � � � i∈[6k] V (Ri) � � \\ {s} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Furthermore, we set E′ := E(T[V ′]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Observe that T[V ′] is a (disjoint) union of 6k paths of length d − 1, where the vertices of the i-th path are r1 i , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , rd i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Observe further that V (T) = V (Ts) ˙∪V ′ and that E(T) = E′ ˙∪ E(Ts) ˙∪ {{s, r1 i } | i ∈ [6k]} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (9) Next, note that the edges of Q can be decomposed into 6k paths, each of length d − 1: There are k vertices of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each vertex v ∈ V (∆) the graph Q contains, by definition, a gadget corresponding to v, the edges of which can be decomposed into 6 paths P 1 v , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , P 6 v of length d − 1 (formally, the fractured graph Q ♯ τ yields precisely this decomposition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' see Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Additionally, for each v ∈ V (∆) and i ∈ [6], the first vertex of P i v is chosen to be vi as depicted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 30 Parameterised and Fine-grained Subgraph Counting, modulo 2 ▶ Definition 50 (γ, γE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We define a function γ : T[V ′] → V (Q) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Recall that T[V ′] is the union 6k paths P ′ j := r1 j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , rd j for j ∈ [6k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Fix any bijection b : [6k] → V (∆) × [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then γ maps P ′ j to P i v, where b(j) = (v, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In particular, we enforce that the first vertices of the paths are mapped onto each other, that is, γ(r1 j) := vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Additionally, we define γE : E′ → E(Q) by mapping e to γ(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Observation 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The function γ is an edge-bijective homomorphism from T[V ′] to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Specifically, γE is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Now let (G, c) be a Q-coloured graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We state the following explicitly, since it will be crucial in our reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Observation 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let (G, c) be a Q-coloured graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The mapping cE ◦ γ−1 E is a map from E(G) to E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let us now construct a graph ˆG from a Q-coloured graph G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' an illustration is provided in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Definition 53 (( ˆG, ˆγ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let (G, c) be a Q-coloured graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The graph ˆG is an edge-coloured graph, with colouring ˆγ : E( ˆG) → E(T), constructed as follows: (A) The graph ˆG contains G as a subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each e ∈ E(G) we set ˆγ(e) = γ−1 E (cE(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (B) The vertex set of ˆG is the union of V (G) and V (Ts), and pairs of vertices in V (Ts) are connected by an edge in ˆG if and only they are adjacent in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each such edge e, ˆγ(e) = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (C) The remaining edges of ˆG are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each edge e = {s, r1 j} ∈ E(T), we connect s to all vertices in G that are coloured (by c) with γ(r1 j) (see Definition 50), and for each of those newly added edges e′ we set ˆγ(e′) := e Observe that ˆγ colours the edges of ˆG with E(T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' the cases (A), (B), and (C) correspond, respectively, to the sets E′, E(Ts) and {{s, r1 i } | i ∈ [6k]} (see Equation (9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Similarly to the case of unbounded C-gadgets, for each element Tcol ∈ ColSub(T → ( ˆG, ˆγ)) the induced subgraph Tcol[G] := Tcol[V (Tcol) ∩ V (G)] of Tcol is an edge-colourful subgraph in G, that is, Tcol[G] contains precisely one edge per edge-colour of G under the edge colouring ˆγ hence it contains precisely one edge per edge- colour of G under cE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' As shown in Section 3 in the full version [30] of [31], Tcol[G] thus induces a fracture ρ = ρ(Tcol) of Q: Two edges {v, w} and {v, y} of Q are in the same block in the partition ρv corresponding to vertex v of Q if and only if the edges of Tcol[G] that are coloured γ−1 E ({v, w}) and γ−1 E ({v, y}) are adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In what follows, we show that ρ must always be equal to τ(∆, T, H) (see Definition 48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Lemma 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For every Tcol ∈ ColSub(T → ( ˆG, ˆγ)) we have that ρ(Tcol) = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let Tcol ∈ ColSub(T → ˆG, ˆγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since Tcol must include each of the edge colours given by ˆγ (precisely) once, we have that Tcol must fully contain Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that Ts fully contains T except for 6k rays of length d, and the only way to attach those rays in ˆG is via the vertex s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Now consider the subgraph Tcol[G + s] of Tcol defined as follows: Tcol[G + s] := Tcol[(V (Tcol) ∩ V (G)) ∪ {s}] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since Tcol includes all edge colours given by ˆγ, we have that s must have degree 6k in Tcol[G + s]: By (C) in Definition 53, the vertex s must be connected (within Tcol[G + s]) to one vertex in each of the colour classes Vi = c−1(vi) for v ∈ V (∆) and i ∈ [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Additionally, this implies the following: L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Goldberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Roth 31 Figure 10 The construction of ˆG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The graph G within ˆG is depicted in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 32 Parameterised and Fine-grained Subgraph Counting, modulo 2 ▶ Observation 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Tcol[G + s] is isomorphic to the d-stretch of K1,6k with s at the centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In the remainder of the proof, we will show that the only way for Tcol to (colourfully) embed the 6k rays of length d is as depicted in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that this will conclude the proof since the induced fracture of the depicted embedding is τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Hence we proceed with proving the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We first consider, for each edge {v, x} ∈ E(∆), the vertex vx = (xv) of Q (see Definition 46 and Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The vertex vx has two neighbours nv and nx in Q, where nv denotes the neighbour in the gadget of v and nx denotes the neighbour in the gadget of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Recall that we write Vx = c−1(vx), Nv = c−1(nv), Nx = c−1(nx) ⊆ V (G) for their colour class within G (and thus within ˆG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since Tcol is edge- colourful, it must contain precisely one edge ev between Vx and Nv and one edge ex between Vx and Nx (see (A) in Definition 53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Now observe that every vertex in Vx has distance (at least) d to s within ˆG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This has two crucial consequences: First, the endpoints of ev and ex inside Vx cannot be equal: Otherwise, they could not be part of a ray of length precisely d with source s, and this would contradict the previous observation that Tcol[G + s] is isomorphic to the d-stretch of K1,6k with s at the centre (Observation 55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Hence, second, the endpoints of ev and ex inside Vx both have degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Consequently, they must be the endpoints of two of the rays of length d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' However, the only way for this to be true is them each being connected to s as depicted in Figure 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' in all other cases, Tcol[G + s] cannot be isomorphic to the d-stretch of K1,6k with s at the centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The second consequence implies that the edge colours corresponding to the edges in the paths P 2 v , P 4 v , and P 6 v are covered for each v (recall that Tcol must include each edge colour precisely once).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus, the only possibility to include the remaining edge colours corresponding to the paths P 1 v , P 3 v , and P 5 v while keeping Tcol[G + s] being isomorphic to the d-stretch of K1,6k, is to embed, for each gadget, the remaining 3 rays of length d as depicted in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ◀ We are now able to prove the main lemma of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Lemma 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ⊕Emb((Q ♯ τ, cτ) → (G, c)) = ⊕ColSub(T → ( ˆG, ˆγ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thanks to Lemma 54, the proof is similar to the proof of Lemma 44: Colour- preserving embeddings in Emb((Q ♯ τ, cτ) → (G, c)) are uniquely identified by their image, and a bijection b from ColSub(T → ( ˆG, ˆγ)) to images of colour-preserving embeddings in Emb((Q ♯ τ, cτ) → (G, c)) is given by b : Tcol �→ Tcol[G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ◀ Similarly to the proof in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1, Lemma 56 is sufficient for hardness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Lemma 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let T be a recursively class of trees of unbounded star number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then ⊕Sub(T ) is ⊕W[1]-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The proof is almost identical to the proof of Lemma 45, with the exception that we use Q, τ, ˆG, and ˆγ as defined in the current section, and that we rely on Lemma 56 for the identity ⊕Emb((Q ♯ τ, cτ) → (G, c)) = ⊕ColSub(T → ( ˆG, ˆγ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The remainder of the proof transfers verbatim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ◀ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Goldberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Roth 33 Figure 11 Illustration of the unique way to colourfully embed T into ˆG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The induced fracture is τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 34 Parameterised and Fine-grained Subgraph Counting, modulo 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='3 Unbounded Fork number We will rely on the same high-level strategy as the one that we used when the C-number or star number was unbounded: Given a tree T with large a-b-fork number, we start with a properly chosen cubic graph ∆, and we construct a graph Q which depends on T and ∆, and which contains ∆ as a minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Afterwards, we show that for any Q-coloured graph (G, c) we can construct an edge-coloured graph ( ˆG, ˆγ) where the co-domain of ˆγ is E(T) such that #ColSub(T → ( ˆG, ˆγ)) is equal (modulo 2) to #Emb((Q ♯ τ, cτ) → (G, c)) for a particular fracture τ of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' However, proving this equality will be more involved than it was in the previous cases: In Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='2, we were able to prove, implicitly, that #ColSub(T → ( ˆG, ˆγ)) = #Emb((Q ♯ τ, cτ) → (G, c)), that is, we were able to establish equality, rather than equality modulo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In the current case, we are not able to prove equality and must therefore rely on parity arguments, which makes the case slightly more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We start by fixing the following: Positive integers k, a and b with a ≤ b and k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A tree T with Fa,b(T) ≥ 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' By definition of forks (Definition 29), T contains designated sources s1 1, s2 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , s1 k, s2 k such that for each (i, j) ∈ [k] × [2], the source sj i is the source of two (distinct) rays Fa(i, j) of length a and Fb(i, j) of length b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Additionally degNL(sj i) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' that the designated sources are ordered by their leaf-degrees, that is degL(s1 1) ≥ degL(s2 1) ≥ · · · ≥ degL(s1 k) ≥ degL(s2 k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (10) Consider Figure 12 for an illustration of T, its designated sources, and the rays Fa(i, j) and Fb(i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A k-vertex bipartite cubic graph ∆ with vertices V (∆) = {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , vk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A proper 3-edge-colouring C : E(∆) → {s, m, ℓ} of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='6 We first note that, since there are at least 2k ≥ 4 sources in T, any pair of distinct sources must not be adjacent: Otherwise, the tree T would either be disconnected, or one of the sources would have degNL at least 2, both of which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Observation 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For any distinct pair (i, j) ̸= (i′, j′) we have that sj i and sj′ i′ are not adjacent in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Next, we define the graph Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Definition 59 (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The graph Q is obtained from ∆ and C via substituting vi by the gadget depicted in Figure 13 for each i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Afterwards, for every edge e = {vi, vj} of ∆ we identify the vertex coloured with C(e) in the gadget of vi with the vertex coloured with C(e) in the gadget of vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' While Definition 59 will be useful in our proofs, we note the following easier equivalent way to define Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Observation 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The graph Q is obtained from ∆ and C by substituting each edge of colour s (of ∆) with a path of length 2a, each edge of colour m with a path of length 2b, and each edge of colour ℓ with a path of length 2(a + b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Consequently, ∆ is a minor of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The fracture τ of Q that we will be interested in is defined as follows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Figure 14 depicts the fractured graph Q ♯ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 6 That is, C(e1) ̸= C(e2) whenever e1 ̸= e2 share a vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that every cubic bipartite graph has a 3-edge-colouring by Hall’s Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Goldberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Roth 35 Figure 12 A tree T with Fa,b(T) ≥ 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that the parents of the sj i are not necessarily distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The rays Fa(i, j) and Fb(i, j) are depicted in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' v1 i v2 i a ℓ b s a m b Figure 13 A vertex gadget in the construction of Q in Definition 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A dashed line labelled with a (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' b) depicts a path of length a (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 36 Parameterised and Fine-grained Subgraph Counting, modulo 2 Figure 14 The fractured graph Q ♯ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that the illustration only depicts the fracturing of a single vertex gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Definition 61 (τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let Q be the graph defined in Definition 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each edge e = {vi, vj} of ∆, there is a vertex C(e) ∈ {s, m, ℓ} of degree 2 that connects the gadgets of vi and vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We let τC(e) be the partition consisting of two singleton blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each vertex vi of ∆, the gadget of vi in Q contains the vertex v1 i of degree 3 which is connected to s via a path of length a, to m via a path of length b, and to ℓ via a path of length a + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let es, em, and eℓ be the first edges on those paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We set τvi = {{es, em}, {eℓ}} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For all other vertices u of Q, we let τu be the partition consisting only of one block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Next we identify specific substructures of T that will be necessary in the construction of ˆG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Definition 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Recall that sj i with (i, j) ∈ [k] × [2] are the designated sources of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' T ′ is the graph obtained from T by deleting, for each (i, j) ∈ [k] × [2], the designated source sj i as well as all rays with source sj i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each (i, j) ∈ [k] × [2], pj i is the neighbour of sj i which is not contained in a ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that pj i is unique by definition of forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that pj i ∈ V (T ′) and that the pj i are not necessarily pairwise distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each (i, j) ∈ [k] × [2], dj i = degL(sj i) − 2, that is, dj i is the number of rays with source sj i minus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that dj i ≥ 0 since each sj i is the source of Fa(i, j) and Fb(i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' F := � (i,j)∈[k]×[2] (Fa(i, j) ∪ Fb(i, j)) , that is, F is the subset of V (T) that contains the vertices of the rays Fa(i, j) and Fb(i, j) (which includes sj i) for each (i, j) ∈ [k] × [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' E′ := E(T[F]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' An illustration of these notions is given in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Goldberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Roth 37 Observe that T[F] is a disjoint union of 2k paths of length a + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Specifically, for each (i, j) ∈ [k] × [2] it contains the path F j i := T[Fa(i, j) ∪ Fb(i, j)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' It turns out that Q is isomorphic to a quotient graph of T[F], since for each vertex vi of ∆, the vertex gadget of vi decomposes into two paths of length a + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In fact, this decomposition is given by the fractured graph Q ♯ τ (see Figure 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Formally, we have the following: ▶ Observation 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' T[F] ∼= Q ♯ τ ∼= 2kPa+b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Similarly to the previous two cases, we introduce functions γ and γE which we will need for defining the edge-colours of ˆG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Definition 64 (γ, γE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We define a function γ : F → V (Q) as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each i ∈ [k], γ maps F 1 i to the (a + b)-path in the gadget of vi from s to m, such that γ(s1 i ) = v1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each i ∈ [k], γ maps F 2 i to the (a + b)-path in the gadget of vi from v1 i to ℓ, such that γ(s2 i ) = v2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Furthermore, we write γE : E′ → E(Q) by setting γE({x, y}) := {γ(x), γ(y)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that the definition of γE is well-defined since γ is a homomorphism by Observation 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Concretely, γ can be viewed as the composition of an isomorphism from T[F] to Q ♯ τ and the Q-colouring cτ of Q ♯ τ (see Definition 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Furthermore, γE is clearly a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Hence, similarly to the previous sections, we point out the following: ▶ Observation 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let (G, c) be a Q-coloured graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The mapping cE ◦ γ−1 E is a map from E(G) to E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We are now able construct a graph ˆG from a Q-coloured graph G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' an illustration is provided in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Definition 66 (( ˆG, ˆγ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let (G, c) be a Q-coloured graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The pair ( ˆG, ˆγ) is an edge-coloured graph constructed as follows, where the co-domain of ˆγ is E(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (A) The graph ˆG contains G as a subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each e ∈ E(G), define ˆγ(e) = γ−1 E (cE(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (B) The vertex set of ˆG is the union of V (G) and V (T) \\ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (C) Pairs of vertices in V (T)\\F are connected by an edge in ˆG if and only if they are adjacent in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each such edge e, we set ˆγ(e) = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (D) The remaining edges of ˆG are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each edge e ∈ E(T) that connects a vertex z ∈ V (T) \\ F to a vertex y ∈ F there are corresponding edges in ˆG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' These edges connect z to all vertices g ∈ V (G) such that c(g) = γ(y) For each such edge e′ in ˆG, ˆγ(e′) = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In (D), the only edges in T connecting z ∈ V (T) \\ F to a vertex y ∈ F satisfy that y is one of the designated sources sj i, and z is either pj i ∈ V (T ′) or z is contained in one of the dj i rays with source sj i that are not Fa(i, j) or Fb(i, j) (see Definition 62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Similarly to the other cases, for each element Tcol ∈ ColSub(T → ( ˆG, ˆγ)) the induced subgraph Tcol[G] := Tcol[V (Tcol) ∩ V (G)] of Tcol is an edge-colourful subgraph in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Also, Tcol[G] induces a fracture ρ = ρ(Tcol) of Q as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' First, recall that G is Q-coloured by c, and that G is contained in ˆG (see (A) in Definition 66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Next note that Tcol[G] is a subgraph of G that contains each edge colour in the image of cE ◦ γ−1 E precisely once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since γE is a bijection from E′ to E(Q), we can thus equivalently view Tcol[G] as a subgraph of G that contains each edge colour in the image of cE precisely once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This fact allows us to define ρ(T ) in terms of the function cE as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 38 Parameterised and Fine-grained Subgraph Counting, modulo 2 Figure 15 The graph ˆG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Depicted in the centre is the part of G (within ˆG) that is coloured with the vertices of the i-th vertex gadget of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Depicted in black are the subtree T ′ of T (left), and, as dashed lines, the inner edges of the d1 i + d2 i rays incident to s1 i and s2 i (right) — here, the inner edges are those that are not incident to the sources s1 i and s2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Each edge of ˆG fully contained in the black part has a unique colour w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ˆγ (see Definition 66 (C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Pairs consisting of remaining edges have the same colour (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ˆγ) if and only if they are depicted with the same colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Goldberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Roth 39 Figure 16 Illustration of the condition that yields invalid trees at (i, 1) (below) and (i, 2) (above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Edges contained in E′ are coloured red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Definition 67 (ρ(Tcol)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let Tcol be an element of ColSub(T → ( ˆG, ˆγ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The fracture ρ = ρ(Tcol) of Q is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Two edges {v, w} and {v, y} of Q are in the same block in the partition ρv corresponding to vertex v of Q if and only if the edges of Tcol[G] that are coloured by cE with {v, w} and {v, y} are incident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' With ( ˆG, ˆγ) defined, we can finally state formally the goal of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Recall that (G, c) is a Q-coloured graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Lemma 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Suppose that |c−1(v)| is odd for each v ∈ V (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then ⊕ColSub(T → ( ˆG, ˆγ)) = ⊕Emb((Q ♯ τ, cτ) → (G, c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The proof requires some additional set-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In particular, we need the condition that |c−1(v)| is odd to deal with the case in which what we call “invalid trees” arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' To this end, recall that V j i = c−1(vj i ) denotes the set of vertices in G that are coloured by c with vj i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since G is a subgraph of ˆG (see Definition 66), we slightly abuse notation and write V j i also for the subset of vertices in ˆG corresponding to V j i in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Definition 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let Tcol ∈ ColSub(T → ( ˆG, ˆγ)) and let (i, j) ∈ [k] × [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We call Tcol invalid at (i, j) if the following two conditions are met: (I) Tcol contains precisely two vertices x and y in V j i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (II) x is adjacent to pj i and not incident in Tcol to any edge coloured with a colour in E′ (see Definition 66 (A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Otherwise Tcol is called valid at (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We call Tcol an invalid tree if there exists a pair (i, j) ∈ [k] × [2] such that Tcol is invalid at (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Otherwise, we call Tcol a valid tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We write ColSubval(T → ( �G, ˆγ)) for the set of all valid Tcol in ColSub(T → ( �G, �γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Consider Figure 16 for an illustration of Definition 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Lemma 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Suppose that |c−1(v)| is odd for each v ∈ V (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then the number of invalid trees Tcol ∈ ColSub(T → ( ˆG, ˆγ)) is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 40 Parameterised and Fine-grained Subgraph Counting, modulo 2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For the proof, given two tuples (i, j) and (i′, j′) in [k] × [2] we write (i′, j′) < (i, j) if (i′, j′) is lexicographically smaller than (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Write T (i, j) for the set of all Tcol ∈ ColSub(T → ( ˆG, ˆγ)) that are invalid at (i, j) but valid on all pairs (i′, j′) < (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We will prove that T (i, j) is even for all (i, j) ∈ [k] × [2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' this is sufficient for the lemma to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Hence fix (i, j), let Tcol ∈ T (i, j), and let x and y be as in Definition 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since V j i = c−1(vj i ) and for j ∈ [2], vj i is a vertex of Q, the assumption in the statement of the lemma implies that |V j i | is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since x and y are distinct vertices in V j i , V j i contains additional vertices other than x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Fix a vertex x′ ∈ V j i \\ {x, y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Obtain T ′ col from Tcol by deleting x (including edges incident to x) and by adding x′ and the edge {x′, u} for every u that was adjacent to x — this is well-defined since x is not incident to any edge coloured with a colour in E′, and by construction of ˆG (see Definition 66 (C) and (D)) whenever {x, u} ∈ E( ˆG) is an edge not coloured with a colour in E′, then {x′, u} ∈ E( ˆG) for every x′ ∈ V j i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Additionally, {x, u} and {x′, u} have the same edge-colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Hence, clearly, T ′ col an edge-colourful subgraph of ˆG that is isomorphic to Tcol (and thus to T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For this reason, we obtain that T ′ col ∈ T (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' More generally, the observation that T ′ col ∈ T (i, j) allows us to define an equivalence relation on T (i, j): Let Tcol and T ′ col be elements of T (i, j), and let x and x′ be the vertices in Tcol and T ′ col that satisfy (II) in Definition 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We set Tcol and T ′ col to be equivalent if and only if one can obtained from the other by switching x with x′ as defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The size of one equivalence class is precisely |V j i | − 1 = |c−1(vj i )| − 1, which is even by the premise of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ◀ For the proof of Lemma 68, we need to establish some facts about rays and 2-paths of elements Tcol ∈ ColSubval(T → ( ˆG, ˆγ)), which are those Tcol ∈ ColSub(T → ( ˆG, ˆγ)) that are valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We encapsulate these facts in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1 The Proof of Lemma 68 We first note that, thanks to Lemma 70, it suffices to prove that #ColSubval(T → ( ˆG, ˆγ)) = #Emb((Q ♯ τ, cτ) → (G, c)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This requires some preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We first fix the following objects (recall the definitions of 2-path, Definition 27 and ray, Definition 28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Tcol is an element of ColSubval(T → ( ˆG, ˆγ)) Tcol[G] is the graph obtained from Tcol[V (Tcol)∩V (G)] with isolated vertices removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (In fact, our proof will show that, for valid trees Tcol ∈ ColSubval(T → ( ˆG, ˆγ)), the induced subgraph Tcol[V (Tcol) ∩ V (G)] cannot have isolated vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' However, at the current point of the proof, it is easiest to just remove them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=') For any positive integer t, Rt is the set of length-t rays in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Pt is the set of length-t 2-paths in T that are not rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For any positive integer t, Rt col is the set of length-t rays in Tcol and Pt col is the set of 2-paths in Tcol that are not rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that |Rt| = |Rt col| and |Pt| = |Pt col| for all t since T and Tcol are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We will also rely on the following notion of external rays and 2-paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Definition 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A 2-path P of Tcol is called external if the following two conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Except for the endpoints, none of the vertices of P is contained in V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' P does not contain an edge of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Goldberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Roth 41 Definition 71 applies whether or not P is a ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The following lemmas establish that all 2-paths of Tcol of length greater than b must be external.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Lemma 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Suppose that t is an integer that is greater than b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Suppose that, for all t′ > t, every 2-path in Rt′ col ∪ Pt′ col is external.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then every 2-path in Rt col ∪ Pt col is external.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We first construct a bijection f from Rt to Rt col.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We will use this bijection to argue that every ray in Rt col is external.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In order to define the bijection, consider a ray R = r0, r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , rt in Rt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since t > b ≥ a, R is not one of the designated rays Fa(i, j) and Fb(i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If r0 is not among the designated sources sj i, then, by the construction of ˆG, R is contained in T ′ and thus R ∈ Rt col.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In this case R is external and we set f(R) := R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Alternatively, suppose that r0 = sj i for some i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then R must be one of the dj i black rays in Figure 12 (see Definition 62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' By the construction of ˆG and the fact that Tcol is edge-colourful, there is a vertex x ∈ V j i such that Tcol contains the path x, r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , rt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In Tcol, as in T, the vertices r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , rt−1 have degree 2 and the vertex rt has degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Vertex x cannot have degree 1 in Tcol since this would disconnect Tcol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Also, vertex x x cannot have degree 2: To see this, assume for contradiction that x has degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then there is an integer t′ > t and a ray R′ ∈ Rt′ col the last vertices of which are x, r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , rt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since x is not an endpoint of the ray and since x ∈ V (G), the ray R′ is not external, contradicting the premise of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Hence x has degree at least 3 and therefore f(R) := x, r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , rt is an external ray of Tcol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The function f is injective by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since Tcol and T are isomorphic, |Rt| = |Rt col| and thus f is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since the image of f only contains external rays, we have shown that every element of Rt col is external.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Every ray in the image of f has the property that its degree-1 endpoint is not contained in V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since the image of f is Rt col, we obtain (∗) Every ray in Rt col has the property that its degree-1 endpoint is not contained in V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' To complete the proof, we show that every 2-path in Pt col is external.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Following the same strategy that we used before, we construct a bijection g from Pt to Pt col.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Every 2-path in the range of g is external, so we will conclude that every element of Pt is external.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In order to define the bijection, consider a 2-path P = p0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , pt in Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If neither of the endpoints of P is among the designated sources sj i, then P is contained in T ′ and thus P ∈ Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In this case, P is external and we set g(P) := P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If exactly one endpoint of P is among the designated sources, say p0 = sj i, then there is a vertex x ∈ V j i such that x, p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , pt is a path in Tcol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The vertices p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , pt−1 have degree 2 in Tcol (as in T) and the vertex pt has degree at least 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If x has degree 1 in Tcol, the ray R = pt, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , p1, x is in Tcol, and its degree-1 endpoint x is in V (G), contradicting (∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Hence x cannot have degree 1 in Tcol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Similarly, x cannot have degree 2, since this would create a 2-path longer than t in Tcol that is not external, which contradicts the premise of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Hence x has degree at least 3, and thus g(P) := x, p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , pt is an external 2-path in Pt col.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For the last case, suppose that both endpoints of P are among the designated sources, say p0 = sj i and pt = sj′ i′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then there are x and y in, respectively, V j i and V j′ i′ such that x, p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , pt−1, y is a path in Tcol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Again, p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , pt−1 must all have degree 2 in Tcol as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We show that both x and y have degree at least 3 in Tcol: If both have degree 1, then Tcol is disconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If one of them has degree 1 and the other one has degree at least 3, then we created a ray of length t whose degree-1 endpoint in in V (G), contradicting (∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If one has degree 1 and the other one has degree 2, then we found a ray longer than t which is not external, contradicting the premise of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If one has degree 2 and the other has degree at least 2, then there is a non-external 2-path longer than t, again 42 Parameterised and Fine-grained Subgraph Counting, modulo 2 contradicting the premise of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus, as desired, both must have degree at least 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Therefore, g(P) := x, p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , pt−1, y is an external 2-path in Pt col.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The function g is injective by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since Tcol and T are isomorphic, |Pt| = |Pt col| and thus g is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since the image of g only contains external 2-paths, we have shown that every element of Pt col is external, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ◀ ▶ Lemma 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Suppose that t is an integer that is greater than b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then every 2-path in Rt col ∪ Pt col is external.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let tmax be the maximum integer for which Rtmax ∪ Ptmax is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let Φt be the proposition “t ≤ b or every 2-path in Rt col ∪ Pt col is external”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We will show by induction on tmax−t that Φt holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The base case arises when tmax−t = 0, so t = tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If tmax ≤ b then Φt is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Otherwise, for each t′ > t, the set Rt′ col ∪ Pt′ col is empty and we can invoke Lemma 72 to conclude that Φt holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For the induction step, consider t such that tmax − t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' By the induction hypothesis, Φt′ holds for all t′ ∈ {t + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , tmax}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If t ≤ b then Φt holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Otherwise, for all t′ > t > b, we know from Φt′ that every 2-path in Rt′ col ∪ Pt′ col is external.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We can then apply Lemma 72 to conclude that every 2-path in Rt col ∪ Pt col is external.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ◀ Before proceeding with the proof of Lemma 68, we provide an overview of the central steps of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Recall that it suffices to prove that #ColSubval(T → ( ˆG, ˆγ)) = #Emb((Q ♯ τ, cτ) → (G, c)) and that we have a fixed an element Tcol of ColSubval(T → ( ˆG, ˆγ)) and proved various properties about it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (1) Our goal is to show that Tcol is embedded in ˆG in the following manner (see Figure 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each (i, j) ∈ [k] × [2], Tcol contains a ray Ra(i, j) of length a and a ray Rb(i, j) of length b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' those rays correspond to the designated rays Fa(i, j) and Fb(i, j) in T (recall that T and Tcol are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=') a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' T ′ is part of Tcol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For every i ∈ [k] and j ∈ [2], the vertices pj i in T ′ is connected to a vertex wj i of G with c(wj i ) = vj i = γ(sj i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In Tcol, the vertex wj i is the source of dj i rays other than Ra(i, j) and Rb(i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The vertices of these dj i rays are not in T ′ and are not in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The edge colours of the edges in these rays in ˆγ are the same as the edge-names in T (see Definition 66 (C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The length-a ray Ra(i, 1) is a path in Tcol from w1 i to the vertex ua(i, 1) of G with some colour c(ua(i, 1)) (a vertex of Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This colour c(ua(i, 1)) corresponds to the vertex “s” in the gadget of the vertex vi of ∆ (see Definition 59 and Figure 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The length-b ray Rb(i, 1) is a path in Tcol from w1 i to the vertex ub(i, 1) of G with some colour c(ub(i, 1)) (a vertex of Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This colour c(ub(i, 1)) corresponds to the vertex “m” in the gadget of the vertex vi of ∆ (see Definition 59 and Figure 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The length-b ray Rb(i, 2) is a path in Tcol from w2 i to the vertex ub(i, 2) of G with some colour c(ub(i, 2)) (a vertex of Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This colour c(ub(i, 2)) corresponds to the vertex “ℓ” in the gadget of the vertex vi of ∆ (see Definition 59 and Figure 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The length-a ray Ra(i, 2) is a path in Tcol from w2 i to the vertex ua(i, 2) ̸= w1 i of G with some colour c(ua(i, 2)) = γ(s1 i ) = v1 i (recall that the colour is a vertex of Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For every edge e = {vi, vi′} in ∆, ua(i, 1) ̸= ua(i′, 1), ub(i, 1) ̸= ub(i′, 1) and ub(i, 2) ̸= ub(i′, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Goldberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Roth 43 (2) We now make some observations about the fracture ρ = ρ(Tcol) from Definition 67, given that Tcol is embedded in ˆG as described in Item (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The definition of Q (Definition 59) tells us that, for every edge e = {vi, vi′} in ∆, there is a degree-2 vertex y of Q that connects the gadgets of vi and vi′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Vertex y corresponds to the vertex C(e) ∈ {s, m, ℓ} in the two gadgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Suppose without loss of generality that C(e) = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The other cases are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' From (1c) the colour C(e) = s is the same as c(ua(i, 1)) and c(ua(i′, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' From (1b) c(w1 i ) = v1 i and c(w1 i′) = v1 i′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since Tcol is colourful and the embedding is as in (1), the edges of the ray from w1 i to ua(i, 1) have different edge colours to the ray from w1 i′ to ua(i′, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus, the edge in G in the first ray that is adjacent to ua(i, 1) (call it ei) has a different colour from the edge n G in the second ray that is adjacent to ua(i′, 1) (call it ei′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Concretely, we have cE(ei) = {s, x} and cE(ei′) = {s, x′} where x and x′ are the neighbours of s (in Q) in the gadgets of vi and vi′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' By (1g) we have ua(i, 1) ̸= ua(i′, 1) and thus, by definition of ρ (Definition 67), ρy consists of two singleton blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Similar arguments show that ρ coincides with τ (see Definition 61) at every vertex of Q that corresponds to vertex “s”, “ℓ” or “m” in any gadget corresponding to any vertex vi of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We now continue with the vertices v1 i for i ∈ [k] of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' See Figure 13 for the gadget containing v1 i in Q and Figure 17 for the graph ˆG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We will use “s”, “ℓ” and “m” as the names of these vertices in the gadget containing v1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The vertex v1 i has degree 3 and is connected to s via a path of length a, to m via a path of length b and to ℓ via a path of length a + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let ys, ym, and yℓ be the successors of v1 i on those paths, that is, the edges incident to v1 i in Q are es := {v1 i , ys}, em := {v1 i , ym}, an eℓ := {v1 i , yℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Now, by (1c) and (1d), the edges of Tcol that are coloured (by cE) with es and em are {w1 i , ra} and {w1 i , rb}, where ra and rb are the successors of w1 i on the rays Ra(i, 1) and Rb(i, 1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Furthermore, by (1f), the edge of Tcol that is coloured (by cE) with eℓ is {ua(i, 2), ˆr} where ˆr is the vertex in the ray Ra(i, 2) that is adjacent to ua(i, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since ua(i, 2) ̸= w1 i (by (1f)), the edge {ua(i, 2), ˆr} is not incident to either {w1 i , ra} or {w1 i , rb}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus ρv1 i = {{es, em}, {eℓ}} which coincides with τv1 i by Definition 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' So τ and ρ coincide at vertex v1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Next are the vertices v2 i for i ∈ [k] (see Figure 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This case is easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If Tcol is embedded as described in (1) (see Figure 17), then, for each i ∈ [k], there is only one vertex of Tcol which is coloured by c with colour v2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This vertex is w2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus every edge of Tcol whose edge colour includes v2 i is incident to w2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Hence ρv2 i only consists of one block, which coincides with τv2 i by Definition 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Finally, every remaining vertex of Q (see Figure 13) has degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let y be such a vertex and let y1 and y2 be the neighbours of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then the edges of Tcol coloured by cE with {y, y1} and {y, y2} must be successive edges on one of the rays Ra(i, 1), Rb(i, 1), Ra(i, 2), or Rb(i, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' So these successive edges are both incident to the vertex of the ray that is coloured y by c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus ρy only consists of one block, which coincides with τy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since we have shown that the fractures ρ and τ coincide at every vertex of Q, we conclude that ρ = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (3) We next explain why it is useful to have ρ = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Recall that our goal is to prove that #ColSubval(T → ( ˆG, ˆγ)) = #Emb((Q ♯ τ, cτ) → (G, c)) and that Tcol is an element of ColSubval(T → ( ˆG, ˆγ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Our method will be to show that the function β defined by 44 Parameterised and Fine-grained Subgraph Counting, modulo 2 β(Tcol) = Tcol[G] is a bijection from ColSubval(T → ( ˆG, ˆγ)) to Emb((Q ♯ τ, cτ) → (G, c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' It will turn out that this implies that the embedding ρ coincides with τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (4) In order to prove Item (1) we will proceed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (i) We show that all 2-paths (including rays) of Tcol are external, except for 2k rays of length b and 2k rays of length a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that we already established this claim for 2-paths of lengths greater than b in Lemma 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (ii) Then we show that Tcol contains two degree-1 vertices in each of the vertex sets L and M of G (within ˆG) — see Figure 17, recalling that, for each vertex gadget, the sets L and M denote the vertex subsets of G that are coloured by c with ℓ and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The point of this is that we will also prove that Tcol has two degree-1 vertices in S (Item 4iv) — this will split off the part of Tcol corresponding to a single gadget, so we will only have to study the embedding of Tcol within each gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We prove the claim about L and M by using the fact that Tcol is isomorphic to T and that all 2-paths longer than b are external.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This implies that if vi and vi′ are the two vertices of ∆ sharing this gadget then the 2-paths between V 2 i and V 2 i′ are covered by two rays in Tcol, both of which end in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (iii) We next show that the degree-1 vertices in (4ii) are the endpoints of 2k rays of length b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We have already seen that for each of the k gadgets the endpoints of these rays are in L and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For the i’th gadget, the sources are in V 1 i and V 2 i If b > a then we show that all remaining 2-paths of length b and also all 2-paths with lengths in a + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , b − 1 are external.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The proof of this claim relies on the same arguments as the proof of Lemma 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (iv) Next, we show that for each gadget, Tcol contains two degree-1 vertices in S — see Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The proof uses the fact that all 2-paths longer than a that are not covered by (4iii) are external.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (v) We next show that the degree-1 vertices in (4iv) are the endpoints of 2k rays of length a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We have already seen that for each of the k gadgets the endpoints of these rays are in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For the i’th gadget, the source is in V 1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (vi) The remaining details of the proof rely on the fact that the tree Tcol is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We now provide the proof in detail;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' for convenience, we also restate the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ▶ Lemma 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Suppose that |c−1(v)| is odd for each v ∈ V (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then ⊕ColSub(T → ( ˆG, ˆγ)) = ⊕Emb((Q ♯ τ, cτ) → (G, c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We will prove that for any Tcol ∈ ColSubval(T → ( ˆG, ˆγ)), Item (1) of the proof overview holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Using this fact and the argument from Item (2) of the proof overview, we conclude that for any Tcol ∈ ColSubval(T → ( ˆG, ˆγ)), ρ(Tcol) = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Recall that every edge-colourful subgraph of G induces a fracture of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let G′ be an element of Emb((Q ♯ τ, cτ) → (G, c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This means that G′ is an edge- colourful subgraph of G that induces τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We wish to see how G′ can be extended to some Tcol ′ ∈ ColSubval(T → ( ˆG, ˆγ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We know from Item (1) that any Tcol ′′ ∈ ColSubval(T → ( ˆG, ˆγ)) can only be embedded in ˆG in one way, so G′ can only be extended in one way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The details are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We claim that there is only one possible extension because T ′ has to be included and item (b) of (1) ensures that, for each j ∈ [2], the vertex pj i is connected to wj i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The rest of (1) shows the unique way to include the rays, so the extension is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let β be the function from ColSubval(T → ( ˆG, ˆγ)) that maps any element Tcol to Tcol[G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that Tcol[G] ∈ Emb((Q ♯ τ, cτ) → (G, c)) since ρ(Tcol) = τ and ρ(Tcol) is a function of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Goldberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Roth 45 Figure 17 An embedding Tcol of T in ˆG that yields the fracture τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We will show that this is the only way to embed T in ˆG in such a way that each edge-colour is used precisely once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that dashed lines depict paths in Tcol, and solid lines depict edges in Tcol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Tcol[G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let β′ be the function that maps an element of Emb((Q ♯ τ, cτ) → (G, c)) to its unique extension in ColSubval(T → ( ˆG, ˆγ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that β◦β′ and β′◦β are both the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Therefore β is a bijection and |ColSubval(T → ( ˆG, ˆγ))| = |Emb((Q ♯ τ, cτ) → (G, c))|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='The lemma follows from Lemma 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' To finish the proof, we will fix Tcol ∈ ColSubval(T → ( ˆG, ˆγ)) and we will show that Item (1) of the proof overview holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Part (a) of (1) is trivial since Tcol is edge-colourful so it contains T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The first sentence of (b) is also trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We will next focus on (c)–(g), noting along the way when the rest of (b) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Recall from Definition 59 that, for each i ∈ [k], the graph Q contains for each vertex vj such that ∆ has an edge e = {vi, vj} with C(e) = m, a path Pi,j of length 2b from v1 i to v1 j , and for each vertex vj such that ∆ has an edge e = {vi, vj} with C(e) = ℓ, a path Pi,j of length 2b from v2 i to v2 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 46 Parameterised and Fine-grained Subgraph Counting, modulo 2 Recall from Definition 6 that cE maps edges of G to edges of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Furthermore, G is a subgraph of ˆG, see Definition 66 (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let Tcol(i, j) be the subgraph of Tcol[G] induced by edges e of G such that cE(e) is in the path Pi,j By construction, Tcol(i, j) is the union of some number of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We will next argue that it is the union of exactly two disjoint length-b paths: If Tcol(i, j) has more than two components then at least one component is disconnected from T ′ in Tcol, contradicting the fact that Tcol is a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If Tcol(i, j) is a single path then it is contained in a 2-path of length at least 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since this 2-path contains an edge in G, it is not external (Definition 71).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This contradicts Lemma 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If Tcol(i, j) is the union of exactly two disjoint paths, one of which has length larger than b then this larger 2-path is contained in a 2-path that is not external contradicting Lemma 73 What we have shown is that T(i, j) consists of two length-b paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For some t ∈ {1, 2}, one of these paths is from V t i and the other is from V t j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' To be more precise and to fix the notation for t = 1, we have now shown that, for each i ∈ [k], Tcol[G] contains a path Rb(i, 1) of length b that starts at a vertex w1 i ∈ V 1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We refer to the other end of this path as ub(i, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The vertex ub(i, 1) has degree 1 and is contained in M (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=', in c−1(m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We next argue that w1 i has degree at least 3 in Tcol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (See Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=') If w1 i has degree 1 in Tcol then Tcol is disconnected, contradicting the fact that it is a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If w1 i has degree 2 in Tcol, then Tcol has a ray of length at least b + 1 that is not external, which is again a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' By the same reasoning, Tcol contains a ray Rb(i, 2) of length b that starts at a vertex w2 i ∈ V 2 i and ends at a vertex ub(i, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The ray Rb(i, 2) is contained in Tcol[G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We have just finished parts (d) and (e) of (1) and the part of (g) that concerns length b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' So what we have shown corresponds to Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We would now like to prove parts (c) and (f) but unfortunately these are more difficult because we have to show where the rays with lengths between a and b are embedded so that we can argue about where the length-a rays are embedded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Define ˆR := �k i=1{Rb(i, 1), Rb(i, 2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Recall that k, a, and b are positive integers with a ≤ b and k ≥ 2 and that T has Fa,b(T) ≥ 2k and Tcol ∼= T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Also, Rb col is the set of length-b rays in Tcol and Rb is the set of length-b rays in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (See Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=') Using the notation that we have established, we will prove the following claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Claim 1: Let P ∈ (Rb col \\ ˆR) ∪ Pb col.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If a < b then P is external.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We prove Claim 1 for the case where P ∈ Rb col \\ ˆR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The other case is similar but easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Observe that |Rb| ≥ 2k since Fa,b(T) ≥ 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' So Rb can be partitioned as follows Rb[S] is the set of the 2k length-b rays Fb(i, j) whose sources are s1 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , s2 k and which are depicted as red dashed lines in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Rb[T] = Rb \\ Rb[S] contains the remaining rays of length b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Our goal is to show that all rays in Rb col \\ ˆR are external.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' To do this, we first show that |Rb[T]| = |Rb col \\ ˆR| and we then provide an injection from Rb[T] to Rb col \\ ˆR in which all elements of the range are external rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' To show that |Rb[T]| = |Rb col \\ ˆR|, first note that |Rb| = |Rb col| because T and Tcol are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We further have |Rb[S]| = | ˆR| = 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We next define the (injective) map from Rb[T] to Rb col\\ ˆR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For any ray R = r0, r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , rb ∈ Rb[T] we proceed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Goldberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Roth 47 Figure 18 Illustration of the embedding of Tcol after the rays of length b are analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Solid lines depict edges, dashed lines depict paths, and dash-dotted lines depict sequences of edges (the identification of the endpoints of which we have not yet been determined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that both Rb(i, 1) and Rb(i, 2) must be of length b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Except for those two rays, the identification of endpoints of the remaining edges that are incident to G (within ˆG) has not been determined yet either;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' this is depicted by the dotted circles inside the colour classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The fracture ρ induced by Tcol will depend on the identification of the edges of Tcol, both endpoints of which lie in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The goal is to show that the endpoints have to be identified precisely as depicted in Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 48 Parameterised and Fine-grained Subgraph Counting, modulo 2 If r0 is not among the designated sources sj i, then R is fully contained in T ′ (see Figure 12) and thus R is a ray in Tcol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We map R to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that R is external since it is fully contained in T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Otherwise, r0 = sj i and R is one of the rays depicted as black dashed lines in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since Tcol is edge-colourful, and by construction of ˆG, Tcol contains a path R′ = x, r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , rb where x ∈ V j i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (See Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=') If x has degree 1 in Tcol then Tcol is disconnected, which is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If x has degree 2 in Tcol then Tcol has a non-external ray which is longer than b, which is also a contradiction by Lemma 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus, x has degree at least 3 in Tcol, and R′ is an external ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We map R to R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This concludes the proof of Claim 1 for the case where P ∈ Rb col \\ ˆR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ■ Claim 2: Suppose that there is an integer t′ such that a < t′ < b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Suppose that P ∈ Rt′ col∪Pt′ col.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then P is external.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In order to explain the proof of Claim 2, recall that we have established the following facts about 2-paths in Tcol in Lemma 73 and Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Every 2-path of length greater than b is external.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Every 2-path of length b is either a ray in ˆR or is external.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' With those 2-paths covered, the proof of Claim 2 is analogous to the proof of Lemma 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ■ Using Claims 1 and 2 we will now prove parts (c) and (f) of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For each 2-path whose length is larger than a, we have already shown that it is in ˆR or we have shown that it is external.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In order to prove (c) we will show that, for each edge {vi, vi′} of ∆ with colour s, the sequence of edges in Tcol between V 1 i and V 1 i′ is the union of two disjoint length-a rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This is formalised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that for each edge {vi, vj} of ∆ coloured by the 3-edge-colouring C with s, there is a path Pi,j of length 2a from v1 i to v1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Recall that cE maps edges of G to edges of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We write Tcol(i, j) for the subgraph of Tcol[G] induced by edges e of G such that cE(e) is in the path Pi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' By construction, Tcol(i, j) is the union of some number of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We will next argue that it is the union of exactly two disjoint length-a paths: If Tcol(i, j) has more than two components then at least one component is disconnected from T ′ in Tcol, contradicting the fact that Tcol is a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If Tcol(i, j) is a single path then it is contained in a 2-path of length at least 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since this 2-path contains an edge in G, it is not external (Definition 71).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Additionally, it is not included in ˆR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This contradicts the aforementioned fact that each 2-paths of length at least a + 1 is external or included in the set ˆR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If Tcol(i, j) is the union of exactly two disjoint paths, one of which has length larger than a, then this larger path yields a contradiction similarly to the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' What we have shown is that T(i, j) consists of two length-a paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' One of these paths is from V 1 i and the other is from V 1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' To be more precise and to fix the notation, we have now shown that, for each i ∈ [k], Tcol[G] contains a path Ra(i, 1) of length a that starts at a vertex ˆw1 i ∈ V 1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We refer to the other end of this path as ua(i, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The vertex ua(i, 1) has degree 1 and is contained in S (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=', in c−1(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' So we have established Part (c) of item (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Consider Figure 19 for an illustration of all the information we gathered so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (The vertices labelled zj i and the edge set Ea i in the figure will be discussed below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' To finish the proof of item (1) we will show part (f) and the rest of part (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We take these together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Recall that for every i ∈ [k] there is a path P a i = v1 i , y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , ya−1, v2 i of length a in Q from v1 i to v2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since Tcol is edge-colourful, it includes each of the colours of the edges on this path exactly once — these colours are γ−1 E ({v1 i , y1}),γ−1 E ({y1, y2}), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ,γ−1 E ({ya−1, v2 i }).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Goldberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Roth 49 Figure 19 Depiction of the embedding of Tcol as established after Claim 2 (in the proof of Lemma 68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Solid lines depict edges, dashed lines depict paths, and dash-dotted lines depict sequences of edges (the identification of the endpoints of which has not yet been determined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that we have not yet determined how the endpoints inside of the colour classes V 1 i and V 2 i are identified either;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' this is depicted by the dotted circles inside these colour classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Proving that the embedding of Tcol is as depicted in Figure 17 requires us to show that all endpoints in V 2 i are identified, and that all endpoints in V 1 i , except for x1 i , are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 50 Parameterised and Fine-grained Subgraph Counting, modulo 2 Under the edge colouring cE, the same edges of Tcol are coloured with the colours {v1 i , y1}, {y1, y2}, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , {ya−1, v2 i }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , ea be the edges of Tcol with those colours;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' we write Ea i for this set of edges (as is depicted in Figure 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We let x1 i be the vertex of Tcol which is contained in V 1 i and incident to e1, and we let x2 i be the vertex of Tcol which is contained in V 2 i and incident to ea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let z1 i and z2 i be the vertices of Tcol in V 1 i and V 2 i that are adjacent to p1 i and p2 i — those vertices are depicted in Figure 19 and we point out that, a priori, x1 i might be equal to to z1 i and x2 i might be equal to z2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Claim 3: There are no vertices in V (Tcol) ∩ V 1 i other than z1 i , x1 i , w1 i , ˆw1 i and vertices in the d1 i rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' To prove Claim 3, assume for contradiction that z is such a vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Recall that V 1 i is an independent set (because vertices in V 1 i all receive the same colour under c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=') Since Tcol is connected, z has a neighbour outside of V 1 i but all of the edge colours incident to V 1 i are already used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ■ The proof of the following claim is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Claim 4: There are no vertices in V (Tcol) ∩ V 2 i other than z2 i , x2 i , w2 i , and vertices in the d2 i rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ■ Claim 5: Both z1 i and z2 i have degree at least 3 in Tcol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We prove the claim for z1 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' an analogous argument applies for z2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Assume first for contradiction that z1 i has degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since Tcol is connected, Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='5 implies that |V (Tcol)∩V 1 i | = 2 so x1 i = w1 i = ˆw1 i and the depicted vertices in the d1 i rays are also identified with this vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' By Definition 69, Tcol is invalid, giving a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Now assume for contradiction that z1 i has degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We consider two subcases: z1 i is identical to x1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then Tcol is disconnected, which yields a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' z1 i is identical to w1 i or ˆw1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This is an immediate contradiction since sources cannot have degree 2 (recall that we already established Ra(i, 1) and Rb(i, 2) to be rays).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' zi is incident to the first edges of one of the additional d1 i outgoing paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' However, in this case, Tcol can only be connected if there is precisely one further vertex of Tcol in V 1 i that is incident to all outgoing edges not covered by z1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' However, in this case, Tcol is an invalid tree, yielding the desired contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since the three cases above are exhaustive, the proof of Claim 5 is concluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ■ Next we need the following property: Claim 6: Let t be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If t < a then each ray in Rt col is external.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' For the proof, recall that |Rt| = |Rt col| since T and Tcol are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that each ray R of length t of T is either fully contained in T ′, or it is one of the dj i black rays for some (i, j) ∈ [k] × [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (See Figure 12) If R is fully contained in T ′, then R is also contained in Rt col and it is external.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If R = r0, r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , rt is one of the dj i black rays, then Tcol contains a path R′ = y0, r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , rt for some y0 ∈ V j i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Suppose that y0 has degree at least 3 in Tcol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then, as in Claim 1, R′ is then an external ray, and we are finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We next consider the case where y0 has degree 1 or 2 in Tcol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If the degree is 1, then Tcol is disconnected, leading to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If the degree is 2, then y0 ̸= zj i by Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus, the only way for Tcol not being disconnected is y0 = xj i and Tcol[Ea i ] is a path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' However, then we obtained a ray of length at least a + t which is neither external, nor in the set ˆR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus, we obtain a contradiction by either Claim 2 (a + t < b), or by Claim 1 (a + t = b), or by Lemma 73 (a + t > b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This concludes the proof of Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ■ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Goldberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Roth 51 Next, observe that Tcol cannot connect z1 i and z2 i via a path within G, that is, via a path containing the edges Ea i : Otherwise Tcol would contain a cycle since p1 i and p2 i are connected by a path within T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We will see that z1 i and z2 i are sources of Tcol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let S be the set of all sources of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Consider the multi-set of leaf-degrees of T degL(S) := {{degL(s) | s ∈ S}} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let Scol be the set of all sources of Tcol and let degL(Scol) be the muti-set of leaf-degrees Tcol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since Tcol and T are isomorphic, the multi-sets degL(S) and degL(Scol) are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Suppose that s ∈ S is a source of T not among the designated sources sj i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then s is contained in T ′, and it is also a source of Tcol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since all of the zj i have degree at least 3 in Tcol (by Claim 5), they cannot be part of further rays with source s in Tcol so s has the same leaf-degree in T and in Tcol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We next show that for each i ∈ [k], the set V 1 i ∪ V 2 i contains at least 2 sources of Tcol: Either z1 i is a source or it is connected by a 2-path within Tcol[G] to another source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' However, the only vertices reachable in Tcol[G] from z1 i that can have degree at least 3 are contained in V 2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Similarly, either x2 i is a source or it is connected by a 2-path within Tcol[G] to a source in V 1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We have already seen that z1 i cannot be connected to z2 i within Tcol[G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus the sources reachable from z1 i and z2 i within Tcol[G] must be distinct, and we have shown that for each i ∈ [k], the set V 1 i ∪ V 2 i contains at least 2 sources of Tcol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since Tcol and T have the same number of sources, and since 2k sources of T are not contained in T ′, we have thus shown that for each i ∈ [k], the set V 1 i ∪ V 2 i contains precisely 2 sources of Tcol;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' let us denote those 2 sources by ˆz1 i and ˆz2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Now, consider the following subsets of S and Scol: S′ := {s1 1, s2 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , s1 k, s2 k} is the set of designated sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' S′ col := {ˆz1 1, ˆz2 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' , ˆz1 k, ˆz2 k} is the set of sources of Tcol in G (within ˆG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since we already know that degL(S \\ S′) = degL(Scol \\ S′ col) (those are the sources in T ′), we require degL(S′) = degL(S′ col) for T and Tcol to be isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' What follows is the final claim within the proof of this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Claim 7: For all i ∈ [k], the following five conditions are satisfied: {z1 i , z2 i } = {ˆz1 i , ˆz2 i }, that is, z1 i and z2 i are the two sources in V 1 i ∪ V 2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Tcol contains precisely 2 vertices in V 1 i : One is z1 i and one is x1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' x1 i has degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Further, z1 i , w1 i , ˆw1 i and all the endpoints of the d1 i rays are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Tcol contains precisely 1 vertex in V 2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Further, z2 i , x2 i , w2 i and all endpoints of the d2 i rays are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Tcol[Ea i ] is a ray with source z2 i (= x2 i = w2 i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Before proving Claim 7, we point out that (1b) and (1f) follow immediately from Claim 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' see Figure 17 and observe that Tcol[Ea i ] becomes the ray Ra(i, 2), and x1 i becomes the endpoint ua(i, 2) of Ra(i, 2) for each i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus the proof of this lemma is concluded if Claim 7 is proved, which is done below: We first show that {z1 i , z2 i } = {ˆz1 i , ˆz2 i } for each i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let Φ = � s∈S′ degL(s) and Φcol = � s∈S′ col degL(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Observe that degL(S′) = degL(S′ col) implies Φ = Φcol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We start by observing that degL(ˆz1 i ) + degL(ˆz2 i ) ≤ (d1 i + 2) + (d2 i + 1) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' There are d1 i rays from V 1 i and also Ra(i, 1) and Rb(i, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' There are d2 i rays from V 2 i and also Rb(i, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' There is also Ea i which could form two rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 52 Parameterised and Fine-grained Subgraph Counting, modulo 2 We next show that Ea i cannot form two rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Assume for contradiction that is does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since Tcol is connected, z1 i , w1 i , ˆw1 i , x1 i and all the endpoints of the d1 i rays are identical, and z2 i , w2 i , x2 i and all the endpoints of the d2 i rays are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Now, if Tcol[Ea i ] would be the disjoint union of two rays of length less than a with sources z1 i and z2 i then those rays are non-external rays of length less than a, contradicting Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We have now shown degL(ˆz1 i ) + degL(ˆz2 i ) ≤ (d1 i + 2) + (d2 i + 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' (11) Next, note that by definition of the dj i (see Figure 12), the following holds: (d1 i + 2) + (d2 i + 2) = degL(s1 i ) + degL(s2 i ) (12) We have now shown that degL(ˆz1 i ) + degL(ˆz2 i ) ≤ degL(s1 i ) + degL(s2 i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Finally, we will show that z1 i and z2 i are sources to finish the first bullet point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Consider z1 i , and recall that is has degree at least 3 by Claim 5, and assume for contradic- tion that it is not a source of Tcol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then z1 i = x1 i , and Tcol[Ea i ] is a path, and x2 i is source (since it is the only vertex in V (Tcol) ∩ V 2 i that might have degree at least 3, except for z2 i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that this also implies that z2 i is a source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus {ˆz1 i , ˆz2 i } = {x2 i , z2 i }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In this case, we have degL(ˆz1 i ) + degL(ˆz2 i ) ≤ d2 i + 1 < degL(s1 i ) + degL(s2 i ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Consequently, using (11) and (12), we have Φcol < Φ, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Thus z1 i is a source of Tcol, and a similar argument shows that z2 i is a source of Tcol as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We now prove the remaining items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In what follows, using the previous bulleted item, we can assume that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ˆz1 i = z1 i and ˆz2 i = z2 i for all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' First, recall that we ordered the sj i by their leaf-degrees, that is degL(s1 1) ≥ degL(s2 1) ≥ · · · ≥ degL(s2 k) ≥ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If x1 1 were equal to z1 1, then Tcol can only be connected if there is only one vertex in V 1 1 , that is, all edges incident to V 1 1 are in fact incident to z1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' However, in that case, we have degL(z1 1) = degL(s1 1) + 1 (by construction of ˆG), and thus the multi-sets cannot be equal anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Hence x1 1 ̸= z1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If x1 1 had degree 2, then there would have been a ray of length at least a+1 that originates in V 2 1 (otherwise Tcol would have been disconnected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' However, this ray would neither be external, nor among the rays in ˆR, contradicting either Lemma 73 or the previous sequence of claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Finally, if x1 1 had degree at least 3, then Tcol would have contained more sources than T, which also yields a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This shows that x1 1 has degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' However, this implies that Tcol can only contain one vertex in V 2 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' otherwise Tcol would be disconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Note that we have just proved the remaining items of Claim 7 for i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Additionally, we have shown that degL(z1 1) = degL(s1 1) and degL(z2 1) = degL(s2 1) Hence we can remove those two numbers from the multi-sets and continue recursively with i = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' This concludes the proof of Claim 7, and thus the proof of the overall lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ◀ We are now ready to conclude the case for trees of unbounded fork number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Goldberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Roth 53 ▶ Lemma 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let T be a recursively enumerable class of trees of unbounded fork number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Then ⊕Sub(T ) is ⊕W[1]-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' We proceed similarly to Lemma 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' However, we have to take care of some subtleties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' First, we start with a class C of cubic bipartite graphs of unbounded treewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Next, we wish to rely on Lemma 68 to obtain the identity ⊕Emb((Q ♯ τ, cτ) → (G, c)) = ⊕ColSub(T → ( ˆG, ˆγ)), where τ is the fracture defined in Definition 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Unfortunately, Lemma 68 only yields the above identity if, for each v ∈ V (Q), |c−1(v)| is odd, that is, each colour class of vertices of G has odd cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' However, this property can easily be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let (G′, c′) be the Q-coloured graph obtained from (G, c) by adding to each even colour class one fresh isolated vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Since Q ♯ τ does not have isolated vertices, this operation does not change the number of colour-preserving embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In combination with Lemma 68 we thus obtain ⊕Emb((Q ♯ τ, cτ) → (G, c)) = ⊕Emb((Q ♯ τ, cτ) → (G′, c′)) = ⊕ColSub(T → ( ˆG′, ˆγ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' From here on, we can proceed analogously to the proof of Lemma 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ◀ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='4 The Dichotomy Theorem for Trees We are now able to prove Theorem 5, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=', an exhaustive and explicit parameterised complexity classification for counting trees modulo 2: ▶ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Let T be a recursively enumerable class of trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' If T is matching splittable, then ⊕Sub(T ) is fixed-parameter tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Otherwise ⊕Sub(T ) is ⊕W[1]-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The fixed-parameter tractability result, as well as the fact that ⊕Sub(T ) is always contained in ⊕W[1] were both shown in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Hence, it remains to prove ⊕W[1]-hardness if T is not matching splittable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' By Lemma 32 each class T of trees that is not matching splittable has unbounded C-number, unbounded star number, or unbounded fork number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Finally, each of these three cases yields ⊕W[1]-hardness as established by Lemmas 44, 56, and 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ◀ References 1 Noga Alon, Phuong Dao, Iman Hajirasouliha, Fereydoun Hormozdiari, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Cenk Sahinalp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Biomolecular network motif counting and discovery by color coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Bioinformatics, 24(13):i241– i249, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1093/bioinformatics/btn163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 2 Noga Alon, Raphael Yuster, and Uri Zwick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Color-coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ACM, 42(4):844–856, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1145/210332.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='210337.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 3 Suman K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Bera, Lior Gishboliner, Yevgeny Levanzov, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Seshadhri, and Asaf Shapira.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Counting subgraphs in degenerate graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ACM, 69(3):23:1–23:21, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1145/3520240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 4 Andreas Björklund, Holger Dell, and Thore Husfeldt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The parity of set systems under random restrictions with applications to exponential time problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In Magnús M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Halldórsson, Kazuo Iwama, Naoki Kobayashi, and Bettina Speckmann, editors, Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I, volume 9134 of Lecture Notes in Computer Science, pages 231–242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Springer, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1007/978-3-662-47672-7\\_19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 5 Marco Bressan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Faster algorithms for counting subgraphs in sparse graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Algorithmica, 83(8):2578–2605, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1007/s00453-021-00811-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 54 Parameterised and Fine-grained Subgraph Counting, modulo 2 6 Andrei A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Bulatov and Amirhossein Kazeminia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Complexity classification of counting graph homomorphisms modulo a prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In Stefano Leonardi and Anupam Gupta, editors, STOC ’22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, pages 1024–1037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ACM, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1145/3519935.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='3520075.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 7 Jianer Chen, Benny Chor, Mike Fellows, Xiuzhen Huang, David W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Juedes, Iyad A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Kanj, and Ge Xia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Tight lower bounds for certain parameterized NP-hard problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=', 201(2):216–231, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 8 Jianer Chen, Xiuzhen Huang, Iyad A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Kanj, and Ge Xia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Strong computational lower bounds via parameterized complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=', 72(8):1346–1367, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='jcss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 9 Yijia Chen, Marc Thurley, and Mark Weyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Understanding the Complexity of Induced Subgraph Isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In Proceedings of the 35th International Colloquium on Auto- mata, Languages and Programming (ICALP), pages 587–596.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Springer, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1007/ 978-3-540-70575-8\\_48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 10 Radu Curticapean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Counting matchings of size k is w[1]-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In Fedor V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Fomin, Rusins Freivalds, Marta Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Kwiatkowska, and David Peleg, editors, Automata, Languages, and Programming - 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part I, volume 7965 of Lecture Notes in Computer Science, pages 352–363.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Springer, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1007/978-3-642-39206-1\\_30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 11 Radu Curticapean, Holger Dell, and Thore Husfeldt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Modular counting of subgraphs: Match- ings, matching-splittable graphs, and paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In Petra Mutzel, Rasmus Pagh, and Grzegorz Herman, editors, 29th Annual European Symposium on Algorithms, ESA 2021, September 6-8, 2021, Lisbon, Portugal (Virtual Conference), volume 204 of LIPIcs, pages 34:1–34:17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='4230/LIPIcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='ESA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 12 Radu Curticapean, Holger Dell, and Dániel Marx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Homomorphisms are a good basis for counting small subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 210–223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' ACM, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1145/3055399.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='3055502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 13 Radu Curticapean and Dániel Marx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Complexity of counting subgraphs: Only the boundedness of the vertex-cover number counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 130–139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' IEEE Computer Society, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1109/FOCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 14 Marek Cygan, Fedor V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Parameterized Algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Springer, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1007/978-3-319-21275-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 15 Víctor Dalmau and Peter Jonsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The complexity of counting homomorphisms seen from the other side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Theoret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=', 329(1-3):315–323, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='tcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 16 Julian Dörfler, Marc Roth, Johannes Schmitt, and Philip Wellnitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Counting induced subgraphs: An algebraic approach to #w[1]-hardness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Algorithmica, 84(2):379–404, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1007/ s00453-021-00894-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 17 Rodney G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Downey and Michael R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Fellows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Fundamentals of Parameterized Complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Texts in Computer Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Springer, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1007/978-1-4471-5559-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 18 Jörg Flum and Martin Grohe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The parameterized complexity of counting problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=', 33(4):892–922, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1137/S0097539703427203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 19 Jörg Flum and Martin Grohe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Parameterized Complexity Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Texts in Theoretical Computer Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' An EATCS Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Springer, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1007/3-540-29953-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 20 Martin Grohe and Dániel Marx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' On tree width, bramble size, and expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Theory, Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' B, 99(1):218–228, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='jctb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 21 Daniel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Harvey and David R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Wood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' The treewidth of line graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Theory, Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' B, 132:157–179, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='jctb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 22 Russell Impagliazzo and Ramamohan Paturi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' On the complexity of k-sat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=', 62(2):367–375, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1006/jcss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1727.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Goldberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Roth 55 23 Bart M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Jansen and Dániel Marx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Characterizing the easy-to-find subgraphs from the viewpoint of polynomial-time algorithms, kernels, and turing kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In Piotr Indyk, editor, Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 616–629.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' SIAM, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1137/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='9781611973730.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 24 Oleksii Kuchaiev, Tijana Milenković, Vesna Memišević, Wayne Hayes, and Nataša Pržulj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Topological network alignment uncovers biological function and phylogeny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Journal of the Royal Society Interface, 7(50):1341–1354, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1098/rsif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='0063.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 25 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Milo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Shen-Orr, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Itzkovitz, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Kashtan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Chklovskii, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Alon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Network Motifs: Simple Building Blocks of Complex Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Science, 298(5594):824–827, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='5594.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='824.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 26 Ron Milo, Shalev Itzkovitz, Nadav Kashtan, Reuven Levitt, Shai Shen-Orr, Inbal Ayzenshtat, Michal Sheffer, and Uri Alon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Superfamilies of evolved and designed networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Science, 303(5663):1538–1542, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1089167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 27 Norbert Peyerimhoff, Marc Roth, Johannes Schmitt, Jakob Stix, and Alina Vdovina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Paramet- erized (modular) counting and cayley graph expanders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In Filippo Bonchi and Simon J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Puglisi, editors, 46th International Symposium on Mathematical Foundations of Computer Science, MFCS 2021, August 23-27, 2021, Tallinn, Estonia, volume 202 of LIPIcs, pages 84:1–84:15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='4230/LIPIcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='MFCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 28 Norbert Peyerimhoff, Marc Roth, Johannes Schmitt, Jakob Stix, and Alina Vdovina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Paramet- erized (modular) counting and cayley graph expanders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' CoRR, abs/2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='14596, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' URL: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='org/abs/2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='14596, arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='14596.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 29 Norbert Peyerimhoff, Marc Roth, Johannes Schmitt, Jakob Stix, Alina Vdovina, and Philip Wellnitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Parameterized Counting and Cayley Graph Expanders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=', to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 30 Marc Roth, Johannes Schmitt, and Philip Wellnitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Detecting and Counting Small Subgraphs, and Evaluating a Parameterized Tutte Polynomial: Lower Bounds via Toroidal Grids and Cayley Graph Expanders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' CoRR, abs/2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='03433, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='03433.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 31 Marc Roth, Johannes Schmitt, and Philip Wellnitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Detecting and Counting Small Subgraphs, and Evaluating a Parameterized Tutte Polynomial: Lower Bounds via Toroidal Grids and Cayley Graph Expanders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), volume 198 of LIPIcs, pages 108:1–108:16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='4230/LIPIcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='ICALP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 32 Benjamin Schiller, Sven Jager, Kay Hamacher, and Thorsten Strufe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' StreaM – A Stream-Based Algorithm for Counting Motifs in Dynamic Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In Proceedings of the 2nd International Conference on Algorithms for Computational Biology (AlCoB), pages 53–67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Springer Interna- tional Publishing, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1007/978-3-319-21233-3_5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 33 Seinosuke Toda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' PP is as hard as the polynomial-time hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=', 20(5):865– 877, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1137/0220053.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 34 Ngoc Hieu Tran, Kwok Pui Choi, and Louxin Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Counting motifs in the human interactome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Nature communications, 4(1):1–8, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1038/ncomms3241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 35 Charalampos E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Tsourakakis, Jakub Pachocki, and Michael Mitzenmacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Scalable motif- aware graph clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In Proceedings of the 26th International Conference on World Wide Web (WWW), page 1451–1460, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1145/3038912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='3052653.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' 36 Virginia Vassilevska Williams, Joshua R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Wang, Richard Ryan Williams, and Huacheng Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' Finding four-node subgraphs in triangle time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' In Piotr Indyk, editor, Proceedings of the Twenty- Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 1671–1680.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' SIAM, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='1137/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='9781611973730.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} +page_content='111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfuv2o/content/2301.01696v1.pdf'} diff --git a/59AyT4oBgHgl3EQfQfYL/vector_store/index.faiss b/59AyT4oBgHgl3EQfQfYL/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..211bf838ff3f78d7541becf54c55c752e51f5bb6 --- /dev/null +++ b/59AyT4oBgHgl3EQfQfYL/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:cfbf9fddd8cf2ca4e2b592b056fffa8eedaba438ac38b5be06b78ced4da0441c +size 4653101 diff --git a/59AzT4oBgHgl3EQfvP0x/content/2301.01702v1.pdf b/59AzT4oBgHgl3EQfvP0x/content/2301.01702v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d8a9341b3f3d56ee132894fd1e2c9e183dd4c3a1 --- /dev/null +++ b/59AzT4oBgHgl3EQfvP0x/content/2301.01702v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0b3c5db253d0eb6ac0853404a58fb3d0e08619ada09128cf028b8d361b042382 +size 554198 diff --git a/59AzT4oBgHgl3EQfvP0x/vector_store/index.faiss b/59AzT4oBgHgl3EQfvP0x/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..4986b27b13c79db566e0fca43e544ca58a899dd4 --- /dev/null +++ b/59AzT4oBgHgl3EQfvP0x/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e120672f59679ec45f581b7e0e5d6476d0a92e095185f521c8f3df5542bbc26c +size 3932205 diff --git a/59AzT4oBgHgl3EQfvP0x/vector_store/index.pkl b/59AzT4oBgHgl3EQfvP0x/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..69f04ac25807ea46a0cbdb493b36be8f36b36913 --- /dev/null +++ b/59AzT4oBgHgl3EQfvP0x/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8d3b67baeef0ee1e378f0c86641b9cb2645920ceb2a22346f2dff86325ef89c5 +size 152537 diff --git a/5NE4T4oBgHgl3EQfBQty/content/tmp_files/2301.04850v1.pdf.txt b/5NE4T4oBgHgl3EQfBQty/content/tmp_files/2301.04850v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7cad6e4892ea10f8dc329287f7352cfef2283bfd --- /dev/null +++ b/5NE4T4oBgHgl3EQfBQty/content/tmp_files/2301.04850v1.pdf.txt @@ -0,0 +1,1516 @@ +Understanding Difficulty-based Sample Weighting with +a Universal Difficulty Measure⋆ +Xiaoling Zhou1, Ou Wu�1, Weiyao Zhu1, and Ziyang Liang1 +Center for Applied Mathematics, Tianjin University, China. +{xiaolingzhou,wuou}@tju.edu.cn, +weiyaozhu042@outlook.com, ziyangliang@tju.edu.cn +Abstract. Sample weighting is widely used in deep learning. A large number +of weighting methods essentially utilize the learning difficulty of training sam- +ples to calculate their weights. In this study, this scheme is called difficulty-based +weighting. Two important issues arise when explaining this scheme. First, a uni- +fied difficulty measure that can be theoretically guaranteed for training samples +does not exist. The learning difficulties of the samples are determined by multiple +factors including noise level, imbalance degree, margin, and uncertainty. Never- +theless, existing measures only consider a single factor or in part, but not in their +entirety. Second, a comprehensive theoretical explanation is lacking with respect +to demonstrating why difficulty-based weighting schemes are effective in deep +learning. In this study, we theoretically prove that the generalization error of a +sample can be used as a universal difficulty measure. Furthermore, we provide +formal theoretical justifications on the role of difficulty-based weighting for deep +learning, consequently revealing its positive influences on both the optimization +dynamics and generalization performance of deep models, which is instructive to +existing weighting schemes. +Keywords: Learning difficulty · Generalization error · Sample weighting · Deep +learning interpretability. +1 +Introduction +Treating each training sample unequally improves the learning performance. Two cues +are typically considered in designing the weighting schemes of training samples [1]. +The first cue is the application context of learning tasks. In applications such as medical +diagnosis, samples with high gains/costs are assigned with high weights [2]. The second +cue is the characteristics of the training data. For example, samples with low-confidence +or noisy labels are assigned with low weights. Characteristic-aware weighting has at- +tracted increasing attention owing to its effectiveness and universality [3,4,5]. +Many existing characteristic-aware weighting methods are based on an intrinsic +property of the training samples, i.e., their learning difficulty. The measures for the +samples’ learning difficulty can be roughly divided into five categories. +⋆ This study is supported by NSFC 62076178, TJF 19ZXAZNGX00050, and Zhijiang Fund +2019KB0AB03. +Paper published at ECML PKDD 2022 +arXiv:2301.04850v1 [cs.LG] 12 Jan 2023 + +2 +Xiaoling Zhou et al. +– Prediction-based measures. This category directly uses the loss [3,6,7] or the pre- +dicted probability of the ground truth [4,8] as the difficulty measures. This measure +is simple yet effective and is widely used in various studies [3,4]. Their intention is +that a large loss (a small probability) indicates a large learning difficulty. +– Gradient-based measures. This category applies the loss gradient in the measure- +ment of the samples’ learning difficulty [9,10]. Santiagoa et al. [9] uses the norm +of the loss gradient as the difficulty measure. Their intuition is that the larger the +norm of the gradient, the harder the sample. +– Category proportion-based measures. This category is mainly utilized in imbal- +anced learning [11], where the category proportion measures the samples’ diffi- +culty. People believe that the smaller the proportion of a category, the larger the +learning difficulty of samples in this category [11,12]. +– Margin-based measures. The term “margin” refers to the distance from the sample +to the oracle classification boundary. The motivation is that the smaller the margin, +the larger the difficulty of a sample [13]. +– Uncertainty-based measures. This category uses the uncertainty of a sample to mea- +sure the difficulty. Aguilar et al. [14] identify hard samples based on epistemic un- +certainty and leverage the Bayesian Neural Network [15] to infer it. +Varying difficulty measures have a huge impact on a difficulty-based weighting +strategy. The underlying factors which influence samples’ learning difficulty considered +in the above measures include noise level [6,7], imbalance degree [11,12], margin [13], +and uncertainty [14]. However, each measure only considers a single factor or in part, +and comes from heuristic inspirations but not formal certifications, hindering the appli- +cation scope of the measures. It is desirable to theoretically explore a universal measure +capturing all of the above factors. Based on this measure, the role of difficulty-based +sample weighting can be revealed more concretely. However, neither theoretical nor +empirical investigations have been conducted to investigate a unified measure. +Moreover, despite the empirical success of various difficulty-based weighting meth- +ods, the process of how difficulty-based weighting positively influences the deep learn- +ing models remains unclear. Two recent studies have attempted to investigate the influ- +ence of weights in deep learning. Byrd and Lipton [16] empirically studied the train- +ing of over-parameterized networks with sample weights and found that these sample +weights affect deep learning by influencing the implicit bias of gradient descent-a novel +topic in deep learning interpretability, focusing on why over-parameterized models is +biased toward solutions that generalize well. Existing studies on this topic [17,18,19] +reveal that the direction of the parameters (for linear predictor) and the normalized mar- +gin (for nonlinear predictor) respectively converge to those of a max-margin solution. +Inspired by the finding of Byrd and Lipton [16], Xu et al. [20] dedicated to studying +how the understandings for the implicit bias of gradient descent adjust to the weighted +empirical risk minimization (ERM) setting. They concluded that assigning high weights +to samples with small margins may accelerate optimization. In addition, they estab- +lished a generalization bound for models that implement learning by using sample +weights. However, they only discussed the measurement of difficulty by using one of +the indicators (i.e., margin), resulting in that their conclusion is limited and inaccurate +in some cases. Furthermore, their generalization bound cannot explicitly explain why + +Understanding Difficulty-based Sample Weighting with a Universal Difficulty Measure +3 +hard samples are assigned with large weights in many studies. More analyses based on +a universal difficulty measure are in urgent demand. +In this study, the manner of how the difficulty-based weighting affects the deep +model training is deeply investigated. First, our analyses support that the generalization +error of the training sample can be regarded as a universal difficulty measure for captur- +ing all of the four factors described above. Second, based on this unified measure, we +characterize the role of difficulty-based weighting on the implicit bias of gradient de- +scent, especially for the convergence speed. Third, two new generalization bounds are +constructed to demonstrate the explicit relationship between the sample weights and the +generalization performance. The two bounds illuminate a new explanation for existing +weighting strategies. Our study takes the first step of constructing a formal theory for +difficulty-based sample weighting. In summary, our contributions are threefold. +– We theoretically prove the high relevance of the generalization error with four main +factors influencing the samples’ learning difficulty, further indicating that the gen- +eralization error can be used as a universal difficulty measure. +– We reveal how the difficulty-based sample weighting influences the optimization +dynamics and the generalization performance for deep learning. Our results indi- +cate that assigning high weights on hard samples can not only accelerate the con- +vergence speed but also enhance the generalization performance. +– We bring to light the characteristics of a good set of weights from multiple perspec- +tives to illuminate the deep understanding of numerous weighting strategies. +2 +Preliminaries +2.1 +Description of Symbols +Let X denote the input space and Y a set of classes. We assume that the training and +test samples are drawn i.i.d according to some distributions Dtr and Dte over X × Y. +The training set is denoted as T = {x, y} = {(xi, yi)}n +i=1 that contains n training +samples, where xi denotes the i-th sample’s feature, and yi is the associated label. +Let di and w (di) be the learning difficulty and the difficulty-based weight of xi. The +learning difficulty can be approximated by several values, such as loss, uncertainty and +generalization error which will be explained in Section 3. +The predictor is denoted by f (θ, x) and F = {f (θ, ·) |θ ∈ Θ ⊂ Rd}. For the sake +of notation, we focus on the binary setting yi ∈ {−1, 1} with f (θ, x) ∈ R. The sign +of the model’s output f (θ, xi) is the predicted label. However, as to be clarified later, +our results can be easily extended to the multi-class setting where yi ∈ {1, 2, · · · , C}. +For multi-class setting, the softmax function is used to get the probability, and the log- +its are given by {fyj (θ, x)}C +j=1. Given a non-negative loss ℓ and a classifier f (θ, ·), +the empirical risk can be expressed as L(θ, w) = 1 +n +�n +i=1 w (di) · ℓ (yif (θ, xi)). We +focus particularly on the exponential loss ℓ (u) = exp (−u) and logistic loss ℓ (u) = +log (1 + exp (−u)). Let ∇l(u) be the loss gradient and f (x|T) is the trained model on +T. The margin is denoted as γi(T) = yif (θ, xi|T) for the binary setting, where it is +equivalently denoted as γi(T) = fyi (θ, xi|T) − maxi̸=j fyj (θ, xi|T) for the multi- +class setting. + +4 +Xiaoling Zhou et al. +2.2 +Definition of the Generalization Error +Bias-variance tradeoff is a basic theory for the qualitative analysis of the generalization +error [22]. This tradeoff is initially constructed via regression and mean square error, +which is given by +Err = Ex,yET [||y − f(x|T)||2 +2] +≈ Ex,y[||y − f(x)||2 +2] +� +�� +� +Bias ++ Ex,yET [||f(x|T) − f(x)||2 +2] +� +�� +� +V ariance +, +(1) +where f (x) = ET [f (x|T)]. Similarly, we define the generalization error of a single +sample xi as +erri = ET [ℓ (f (xi|T) , yi)] ≈ B (xi) + V (xi) , +(2) +where B (xi) and V (xi) are the bias and variance of xi. +2.3 +Conditions and Definitions +Our theoretical analyses rely on the implicit bias of gradient descent. The gradient de- +scent process is denoted as +θt+1 (w) = θt (w) − ηt∇L (θt [w(d [t])]) , +(3) +where ηt is the learning rate which can be a constant or step-independent, ∇L (θt [w(d [t])]) +is the gradient of L, and w(d [t]) is the difficulty-based weight of difficulty d at time +t. The weight may be dynamic with respect to time t if difficulty measures, such as +loss [3] and predicted probability [4], are used. To guarantee the convergence of the +gradient descent, two conditions following the most recent study [20] are shown below. +– The loss ℓ has an exponential tail whose definition is shown in the supplementary +file. Thus, limu→∞ ℓ(−u) = limu→∞ ∇ℓ(−u) = 0. +– The predictor f(θ, x) is α-homogeneous such that f(c·θ, x) = cαf(θ, x), ∀c > 0. +It is easy to verify that losses including the exponential loss, log loss, and cross-entropy +loss satisfy the first condition. The second condition implies that the activation functions +are homogeneous such as ReLU and LeakyReLU, and bias terms are disallowed. In +addition, we need certain regularities from f(θ, x) to ensure the existence of critical +points and the convergence of gradient descent: +– For ∀x∈X, f(θ, x) is β-smooth and l-Lipschitz on Rd. +The third condition is a common technical assumption whose practical implications are +discussed in the supplementary file. +The generalization performance of deep learning models is measured by the gener- +alization error of the test set ˆL (f) [21], defined as +ˆL (f) = P(x,y)∼Dte[γ(f (x, y)) ≤ 0]. +(4) + +Understanding Difficulty-based Sample Weighting with a Universal Difficulty Measure +5 +0 +50 +100 +150 +200 +0 +1 +2 +3 +4 +5 +Error +Id +0 +100 +200 +300 +400 +-2 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +Error +Id +Noise +Clean +1 (Largest) +2 +3 +4 +5 +6 +7 +8 +9 +10 (Smallest) +1 (Largest) +2 +3 +4 +5 +6 +7 +8 +9 +10 (Smallest) +Fig. 1. (a) Generalization errors of clean and noisy samples on noisy data. The noise ratio is 10% +(b) Generalization errors of samples in ten categories on imbalanced data. The imbalance ratio is +10:1. CIFAR10 and ResNet32 are used. Other values of noise ratio and imbalance ratio following +Ref. [25] are also experimented with and the same conclusions can be obtained. +2.4 +Experiment Setup +Demonstrated experiments are performed to support our theoretical analyses. For the +simulated data, the linear predictor is a regular regression model, and the nonlinear pre- +dictor is a two-layer MLP with five hidden units and ReLU as the activation function. +Exponential loss and standard normal initialization are utilized. CIFAR10 [23] is exper- +imented with, and ResNet32 [24] is adopted as the baseline model. For the imbalanced +data, the imbalance setting follows Ref. [11]. For the noisy data, uniform and flip label +noises are used and the noise setting follows Ref. [25]. The models are trained with a +gradient descent by using 0.1 as the learning rate. +The model uncertainty is approximated by the predictive variance of five predic- +tions. To approximate the generalization error, we adopt the five-fold cross-validation [26] +to calculate the average learning error for each sample. +3 +A Universal Difficulty Measure +As previously stated, four factors pointed out by existing studies, namely, noise, imbal- +ance, margin, and uncertainty, greatly impact the learning difficulty of samples. Nev- +ertheless, existing measures only consider one or part of them, and their conclusions +are based on heuristic inspirations and empirical observations. In this section, we theo- +retically prove that the generalization error of samples is a universal difficulty measure +reflecting all four factors. All proofs are presented in the supplementary file. Without +increasing the ambiguity, the generalization error of the samples is termed as error for +brevity. +3.1 +Noise Factor +Noise refers to data that is inaccurate in describing the scene. Numerous studies devoted +to reducing the influence of noisy samples in the dataset on the deep learning models + +6 +Xiaoling Zhou et al. +and these literature intuitively consider noisy samples as hard ones without formal cer- +tification [7,27]. The two kinds of noise are feature noise [31] and label noise [27]. We +offer two propositions to reveal the relationship between the generalization error and +the noise factor. For feature noise, we offer the following proposition: +Proposition 1. Let ∆xi be the perturbation of sample (xi, yi), which is extremely +small in that o(∆xi) can be omitted. Let ∠ϕ be the angle between the direction of +∆xi and the direction of ET [f ′ (xi|T)]. If ET [f ′ (xi|T) · ∆xi] < 0 (i.e., ∠ϕ > 90◦), +then the error of the noisy sample is increased relative to the clean one. Alternatively, +the direction of the perturbation ∆xi and that of ET [f ′ (xi|T)] are contradictory. Oth- +erwise, if ET [f ′ (xi|T) · ∆xi] > 0, then ∠ϕ < 90◦, and the error of the noisy sample +is decreased. +According to Proposition 1, feature noise can be divided into two categories, which +increase or decrease the learning difficulty (generalization error) of the samples, respec- +tively. In this paper, noise that increases the error is called the adversarial type, which is +always used in the field of adversarial learning; otherwise, it is a promoted type, which +refers to noise that decrease the learning difficulty of samples. Therefore, the variation +of the error under feature noise is determined by the noise type. For example, as all +feature noises are adversarial in adversarial learning [32], all of the samples’ errors are +increased with feature noise. For label noise, we offer the following proposition: +Proposition 2. Let π be the label corruption rate (i.e., the probability of each label +flipping to another one). Denote the probability of correct classification for the original +samples as p. If p > 0.5, then the errors of the noisy samples are larger than those of +the clean ones. +This proposition implies that the errors of the samples with label noises are larger +than those of the clean ones on the average. Specifically, if a sample is more likely to be +predicted correctly, its generalization error is increased due to label noise. Let L∗ be the +global optimum of the generalization error of the clean dataset and y′ be the corrupted +label. When the noise in Proposition 2 is added, the empirical error L′ is +L′ = (1 − π) L∗ + πL (f (x) , y′) , +(5) +where we have taken expectations over the noise. When π → 0, the noise disappears, +and the optimal generalization is attained. Proposition 2 is consistent with the empirical +observation shown in Fig. 1(a), where the noisy samples have larger errors than the +clean ones on the average. +3.2 +Imbalance Factor +Besides noise, imbalance is another common deviation of real world datasets. The cat- +egory distribution of the samples in the training set is non-uniform. Various methods +solve this issue by assigning high weights on samples in tail categories which are con- +sidered to be hard ones [4,11]. Nevertheless, a theoretical justification about why these +samples are harder lacks. The imbalance ratio is denoted by cr =max{c1, c2, · · · , cC}: +min{c1, c2, · · · , cC}. Then, we offer the following proposition. + +Understanding Difficulty-based Sample Weighting with a Universal Difficulty Measure +7 +Fig. 2. (a) Correlation between generalization error and average margin. (b) Correlation between +generalization error and epistemic uncertainty. CIFAR10 and ResNet32 are used in this experi- +ment. All values are normalized. +Proposition 3. If a predictor on an imbalanced dataset (cr > e : 1) is an approximate +Bayesian optimal classifier (as the exponential loss is an approximation for the zero- +one loss), which is to minimize the total risk, then the average probability of the ground +truth of the samples in the large category is greater than that of the samples in the small +category. +With Proposition 3, it is easy to obtain Proposition A.1 shown in the supplemen- +tary file that the average error of samples in the small category is larger than that of +the samples in the large category, indicating there are more hard samples in the small +category. This proposition is verified by the experiments, as shown in Fig. 1(b). The +tail categories contain more samples with larger errors. To enhance the performance +of the classification model, samples with larger errors should be assigned with higher +weights, as most methods do [11]. Further experiments in Section 5 (Fig. 6) indicate +that the classification performance of the small category can be improved by increasing +its sample weights. +3.3 +Margin Factor +The samples’ margins measure the distances of the samples from the decision boundary. +Some literature intuitively consider a small margin indicates a large learning difficulty +and corresponds to a low confidence of the prediction [33,13]. However, a formal justi- +fication is lacking. We offer the following proposition. +Proposition 4. Let µi be the true margin of xi corresponding to the oracle decision +boundary. The condition is that the functional margins of a sample trained on random +datasets obey a Gaussian distribution. In other words, for sample xi, its functional +margin γi obey a Gaussian distribution N(µi, σ2 +i ). For sample xj, γj ∼ N(µj, σ2 +j ). + +rwr! +www +hEULOL(p)WIDIDMA +wypV +ELLOL +igsMELLOLELLOLbI8 +Xiaoling Zhou et al. +Fig. 3. The distributions of samples’ margins. +when the margin variances of the two samples are same (i.e., σ2 +i = σ2 +j ), if µi ≤ µj, +then erri ≥errj. Similarly, when the true margins of the two samples are the same (i.e., +µi =µj), if σ2 +i ≥σ2 +j , then erri ≥errj. +Proposition 5 indicates a fact that even a sample with a large true margin, as long +as the margin variance is large, it may also have a high learning difficulty. Specifically, +the true margin (i.e., the mean of the functional margin distribution) of a sample and +error are negatively correlated when the margin variances of the samples are equal. By +contrast, the margin variance and error are positively correlated when the true margins +are equal. This illumination revises the current wisdom. The conclusion in which sam- +ples close to the oracle decision boundary are hard ones [20] is not completely correct. +Indeed, the relation between the margin and error of sample xi conforms with the fol- +lowing formula: +erri = ET [e−γi(T )] = e−µi+ 1 +2 σ2 +i , +(6) +where erri, µi, and σi refer to the generalization error, the true margin, and the margin +variance of sample xi, respectively. For the two samples xi and xj, if µi < µj and +σ2 +i < σ2 +j , then we cannot judge whether erri is greater than errj. As shown in Fig. 2(a), +the average margin and error are negatively correlated for most samples, but it is not +absolute, which accords with the above analyses. Although it is intuitive that the func- +tional margin trained on random datasets obeys a Gaussian distribution, we evaluate it +via the Z-scores of the distributions’ Kurtosis and Skewness [34] which is shown in +Fig 3. More margin distribution curves and all Z-score values of the distributions are +shown in the supplementary file. As all Z-scores are in [−1.96, 1.96], under the test +level of α = 0.05, the distribution of margin obeys the Gaussian distribution. +3.4 +Uncertainty Factor +Uncertainties [37] in deep learning are classified into two types. The first type is aleatoric +uncertainty (data uncertainty), which is caused by the noise in the observation data. Its +correlation with the error has been discussed in Section 3.1. The second type is epis- +temic uncertainty (model uncertainty). It is used to indicate the consistency of multiple +predictions. We give the analyses of the relationship between the generalization error +and epistemic uncertainty. +Let T be a training set, and let P(θ|T) be the distribution of the training models +based on T. The predictive variance V ar(f(xi|θ1), · · · , f(xi|θK)) plus a precision + +'00000000 +T0000000 +S0000000 +30000000 +40000000 +20000000 +0 +JO- +ELGUdIGUC2 +50- +30rtigsM +T0000000 +S0000000 +30000000 +40000000 +20000000 +0 +JO- +ELGUdGUcA +30- +30T2000000 +S0000000 +32000000 +30000000 +32000000 +0000000 +42000000 +0 +JO- +ELGUdnIGUCA +30- +30- +如-30000000 +35000000 +34000000 +3Q000000 +ELGdtGUcA +JO- +J2- +30- +32-tigsM +SS000000 +000000ES. +54000000 +32000000 +S2000000 +S3000000 +58000000 +0 +JO- +ELedGUcA +30- +30rtigrsM +55000000 +53000000 +Q000002S +5Q000000 +000000TS. +58000000 +JO- +50- +30-30000000 +35000000 +34000000 +3Q000000 +2- +J O- +J2- +50- +32-WSa +12000000 +50000000 +00000025. +QQQQQQ0E. +32000000 +Q000000. +JO- +5O +30- +40-rtigisM +J0000000 +50000000 +30000000 +40000000 +20000000 +10- +50- +30- +40-WS.a +00000000 +J0000000 +30000000 +30000000 +40000000 +20000000 +JO +5O +30rigrsM +00000000 +J0000000 +50000000 +30000000 +40000000 +20000000 +J +JO- +50- +30- +40-rtigrsM +30000000 +32000000 +30000000 +32000000 +40000000 +42000000 +JO +J2- +30-rtigisM +J0000000 +50000000 +30000000 +40000000 +20000000 +J O +J2- +30ntigrsM +T0000000 +S0000000 +30000000 +40000000 +20000000 +2- +ELGUdGUCA +10- +J2- +30-S0000000 +$2000000 +30000000 +32000000 +0000000 +42000000 +ELGUdGUc2 +10- +J2- +30-tigisM +'00000000 +T0000000 +S0000000 +30000000 +40000000 +20000000 +0 +JO- +ELGUdGUcA +30- +30- +如-Understanding Difficulty-based Sample Weighting with a Universal Difficulty Measure +9 +constant is a typical manner of estimating epistemic uncertainty [35,36]. Take the mean +square loss as an example1, the epistemic uncertainty is +� +Var [xi] :=τ −1 + +1 +|K| +� +k f(xi|θk)⊺f(xi|θk) − E[f(xi|θk)]⊺E[f(xi|θk)], +(7) +where τ is a constant. The second term on the right side of Eq. (7) is the second raw mo- +ment of the predictive distribution and the third term is the square of the first moment. +When K → ∞ and the constant term is ignored, Eq. (7) becomes +� +Var [xi] := +� +θ +||f(xi|θ) − f(xi)||2 +2dP(θ|T). +(8) +If P(θ|T) is approximated by the distribution of learned models on random training sets +which conform to the Gaussian distribution N(T, δI), Eq. (8) is exactly the variance +term of the error defined in Eq. (2) when the mean square loss is utilized. +As the bias term in the error can capture the aleatoric uncertainty and the variance +term captures the epistemic uncertainty, the overall relationship between uncertainty +and error is positively correlated. Nevertheless, the relationship between epistemic un- +certainty and error is not simply positively or negatively correlated. For some samples +with heavy noises, their epistemic uncertainties will be small as their predictions remain +erroneous. However, their errors are large due to their large bias. This phenomenon is +consistent with the experimental results shown in Fig. 2(b). Epistemic uncertainty and +error are positively correlated for some samples, and the two variables are negatively +correlated for other samples. +3.5 +Discussion about Generalization Error +The commonly used difficulty measures, such as loss [3] and gradient norm [9], are +mainly related to the bias term. Shin et al. [27] emphasized that only using loss as the +measurement cannot distinguish clean and noisy samples, especially for uniform la- +bel noise. There are also a few existing studies that use variance [28,29]. For instance, +Agarwal et al. [30] applied the variance of gradient norms as the difficulty measure. +Indeed, both the variance and bias terms should not be underestimated when measur- +ing the samples’ learning difficulty. Our theoretical analyses support that generalization +error including both the two terms can capture four main factors influencing the sam- +ples’ learning difficulty. Thus, the error can be leveraged as a universal measure that +is more reasonable than existing measures. Existing studies generally apply the K-fold +cross-validation method [26] to calculate the generalization error. More efficient error +calculation algorithms are supposed to be proposed which will be our future work. +4 +Role of Difficulty-Based Weighting +This section aims to solve the second issue of explaining the difficulty-based weighting +in deep learning. Based on the universal difficulty measure, the impacts of the difficulty- +based weighting schemes on the optimization dynamics and the generalization perfor- +mance in deep learning are investigated. Compared with the most recent conclusions +1 For other losses, other methods can be used to calculate the predictive variance [26]. + +10 +Xiaoling Zhou et al. +Fig. 4. “Cosine distance” represents the cosine of the angle between the decision boundary (at +that epoch) and the max-margin solution. (a), (b) Cosine distance and average margin of equal +weights and inverse margin weights using the linear predictor. (c), (d) Cosine distance and average +margin of equal weights and inverse margin weights using the nonlinear predictor. (e), (f) Cosine +distance and average margin of equal weights and increasing weights of noisy samples using +the nonlinear predictor on the noisy data. (g), (h) Cosine distance and average margin of equal +weights and increasing weights of samples in tail categories using the linear predictor on the +imbalanced data. More results are placed in the supplementary file. +[20] established only on the margin factor, our theoretical findings, which are based on +our universal measure, are more applicable and precise. +4.1 +Effects on Optimization Dynamics +Linear Predictor We begin with the linear predictors allowing for a more refined +analysis. Xu et al. [20] inferred an upper bound containing the term DKL(p∥w), where +DKL is the Kullback-Leibler divergence and p is the optimal dual coefficient vector. A +smaller value of DKL(p∥w) means that the convergence may be accelerated. There- +fore, to accelerate the convergence, they believe that the weights w should be consistent +with the coefficients p. Alternatively, the samples with small functional margins will +have large coefficients and thus should be assigned with large weights. However, the +functional margin is not the true margin that corresponds to the oracle boundary. There- +fore, their conclusion that samples close to the oracle classification boundary should be +assigned with large weights [20] cannot be well-drawn according to their inference. We +offer a more precise conclusion with the unified difficulty measure (i.e., generalization +error). As before, we assume that the functional margins of a sample xi obey a Gaus- +sian distribution N(µi, σ2 +i ), where µi is the true margin and σ2 +i is the margin variance +of xi. We offer the following proposition: +Proposition 5. For two samples xi and xj, if erri ≥ errj, then we have: +(1) When the optimal dual coefficient pi of xi on a random training set T is a linear +function of its functional margin γi on T, if µi ≤ µj, then ET [pi] ≥ ET [pj] (i.e., +ET [wi] ≥ ET [wj]); + +S(p)Ebocj0'5 - +0°4 - +0.0 +8.0 +I'O -batdgisw Isupg +0 +500 +400 +00 +008 +J000 +0 +500 +400 +e00 +008 +J000 +028.0 +0'S +28.0 + .0 +000.0 +F 0.0 +0a52 +020.0 +8.0 +zre.0 + 0.1 +I000 +oitogrib Ismitqo ot onistaib 2o0 +2nigisM(Ol:) batdgiaw sl baoslsdmi +batdgisw Isupg +0 +500 +400 +e00 +008 +1000 +0 +500 +400 +e00 +800 +J000 +F 0e.0 +F I- +E se.0 +F 0 +I + Ae.0 +Foe.0 +3 +F 80.0 +4 +F 00.1 +oitogrib Ismitqo ot onstaib 2o0 +2nigisMbatdgigw Isupg +0 +500 +400 +e00 +800 +J000 +0 +500 +400 +e00 +800 +J000 +0'2 +5.0- +0.0 +0.0 + s.0 +0'4 - +0.0 +8.0 +8.0 +e.0 + 0. 1 +I'S +0.1 +2nigisMcbatdgigw Isupe +(OI:1) batdgiw 22slo baonslsdmi +0 +500 +400 +eoo +800 +000 +0 +500 +400 +e00 +008 +J000 +-I - +0'4 +0 +2.0 +I - +0.0 +5- +3 - +8.0 +e.0 +4 - +2 1 +noitogrib Ismitqo ot gonstaib 2o0 +2nigisM(s) Eboc +(p) Ebocj +10 +S00 +400 +00a +008 +1000 +0 +S00 +400 +e00 +800 +1000 +batdgigw Isupg +batdgigw Isupg +08e.0 +0'4 - +280.0 +0.0 +0Qe.0 +8.0 +0002 - +I'0 +000.1 +goib zo +2nigisM(c) Ebocj +(g) Ebocj +10 +500 +400 +e00 +008 +1000 +500 +400 +000 +008 +1000 +0'3 +batdgigw Isupg +-I - +badgig Isupg + +.0 +(OI:1) batdgigw 2esl baonslsdmi +(0I: I) batdgigw 2slo baoslsdmi +0 - +0'2 +I - + 0.0 +5- +7.0 + 8.0 +3 - + e.0 +4 - +I'0 - +2 - +gosib izo +2nigisM(3) Ebocj +() Ebocj +:0 +S00 +400 +e00 +008 +1000 +10 +S00 +400 +00a +008 +J000 +08.0 +batdgisw Isupg +0°4 +batdgigw ionl + 2.0 + 28.0 +F 0.0 +0e.0 + 8.0 +- e.0 +F 2e.0 +F 0. 1 +I"I +F 00. 1 +I'S. +ib io +2nigisM(a) Ebocj +(p) Ebocj +0: +S00 +400 +e00 +800 +1000 +0: +S00 +400 +00a +008 +J000 + 2r.0 + 0.0 +batdgigw Isupg +batdgigw ionl +F 08.0 + 5.0 + .0 + 28.0 +F 0.0 +F 0.0 + 8.0 +F 2e.0 +F 0.1 +F 00.1 +上 s.1 +ib io +2nigisMEbocp(α)((L.Ebocp +Ebocp +0 +S00 +400 +000 +800 +1000 +10 +S00 +400 +e00 +800 +1000 +F0 +batdgigw Isupg +batdgigw Isupe +88.0 +(1:1) batdgigw 2esl baoslsdmi +I +(01:1) batdgigw 2eslo baoslsdmi + 0e.0 +5- +F se.0 +3 +F Ae.0 +4 +F ae.0 +2 - +F80.0 +I00 - +ostaib nizo +2nigisMEbocp +Ebocp +10 +500 +400 +e00 +008 +1000 +10 +500 +400 +e00 +008 +1000 +0'2 + 2r.0 +batdgigw Isupg +batdgiow Isup +-0°20 +0.0 +-0'52 - +0' + 00.0 +0'2 +8.0 + 02.0 + e.0 + zr.0 +I'00 - +I'0 - +oib izo +2nigisMbatdgigw Isupg +0 +500 +400 +e00 +800 +1000 +0 +S00 +400 +e00 +800 +1000 +88.0 ++ +.0 + 0e.0 +F 0.0 +F se.0 + 8.0 +F e.0 +F 0.1 +0e.0 +I'S +8e.0 +I'4 +F 00.1 +g01stzib 200 +2nigisMUnderstanding Difficulty-based Sample Weighting with a Universal Difficulty Measure +11 +Fig. 5. (a)-(c) Normalized margin of increasing the weights of noisy samples/samples with small +margins/samples in tail categories. CIFAR10 data is used. Uniform label noise is adopted. The +noise ratio and imbalance ratio are 10% and 10:1. (d) Generalization error of the test set when +the nonlinear model is trained with different weights on simulated imbalanced data with the +imbalance ratio as 10:1. Other noise and imbalance settings are also experimented with and the +same conclusions can be obtained. +(2) When the optimal dual coefficient pi of xi on a random training set T is a +natural exponential function of its functional margin γi on T, ET [pi] ≥ ET [pj] (i.e., +ET [wi] ≥ ET [wj]) always holds. Notably, even when µi > µj, ET [pi] > ET [pj] may +still hold. +The proof is presented in the supplementary file. ET [pi] > ET [pj] implies that +wi > wj holds on the average. The conclusion that samples with small true margins +should be assigned with large weights may not hold on some training sets when pi is +not a linear function of γi [17]. A sample with a small true margin may have a smaller +weight than a sample with a large true margin yet a large error. Thus, a more general +conclusion when pi is not a linear function of γi is that increasing the weights of hard +samples (samples with large generalization errors) may accelerate the convergence, +rather than just for samples with small margins. Other factors, including noise, imbal- +ance, and uncertainty also affect samples’ learning difficulty. Notably, the weights of +the hard samples should not be excessively increased, as to be explained in the succeed- +ing section. We reasonably increase the weights of the hard samples shown in Figs. 4 +and A-3 in the supplementary file indicating that the optimization is accelerated. +We also prove that difficulty-based weights do not change the convergence direction +to the max-margin solution shown in Theorem A.1 in the supplementary file. As shown +in Fig. 3, the cosine distance and margin value are always increasing during the training +procedure, indicating the direction of the asymptotic margin is the max-margin solution. +Nonlinear Predictor Analyzing the gradient dynamics of the nonlinear predictors is +insurmountable. The main conclusion obtained by Xu et al. [20] can also be established +for difficulty-based weights only if the bound of weights is larger than zero. However, +their theorem has only been proven for binary cases as the employed loss is inapplicable +in multi-class cases. Here, we extend the theory to the multi-class setting with a regu- +larization λ||θ||r on the cross-entropy loss. Let θλ (w)∈arg min Lλ (θ, w). Formally, +the dynamic regime for the nonlinear predictor can be described as follows: +Theorem 1. Let w ∈ [b, B]n. Denote the optimal normalized margin as +γ∗ = +max +∥θ(w)∥≤1 min +i (fyi(θ(w), xi) − max +j̸=i (fyj(θ(w), xi))) +(9) + +Ebocp +0 +500 +400 +00 +800 +J000 +0.0 +CE +5.0 +04 +nigisM +a.0 +8.0 +0.1Ebocp +0 +52 +0 +J00 +500 +0.0 +o'S +04 +migisM +a.0 +8.0 +CE +0.1 +O12G +Icieg2u e Meia2 ot Jo12a 2bje2Ebocp2 +0 +J2 +J00 +J52 +J20 +r +500 +- +2.0 +0.1 +2.1 +0.5 +01:1 = gigw +2:I = dgigw +52 +Ismrronl +2 TEbocpEbocp +0 +J00 +500 +300 +400 +200 +e00 +0.0 +CE +0'4 +nigsM +a.0 +8.0 +0.19PCEbocp +0 +52 +20 +J00 +cr1 +500 +0.0 +O'S +0'4 +nigisM +0.0 +8.0 +CE +0.1Ebocp +0 +02 +J00 +J20 +S00 +0.0 +5.0 +migisM +04 +a.0 +M +8.0 +CE +V0126 +0.1Ebocp +0 +J00 +J20 +r +500 +0.0 +0'5 +nigisM +04 +a.0 +8.0 +CE +IpgJSUcG +0.1Ebocp +0 +5O +40 +80 +J00 +JSO +J40 +2.0 +a.0 +VOSIUOA +『.0 +WM +8.0 + e.0Ebocp +0 +40 +80 +J00 +JSO +J40 +500.0 +0°004 +00.0 +2201 +800.0 +010.0 +O'OJS +0'014Ebocp +0 +SO +40 +eo +80 +J00 +JSO +J40 +0.0 +5.0 +nigisM +04 +0.0 +8.0 +0.1EbocpEbocp +0 +52 +02 +J00 +s +J20 +r +500 +2.0 +0.I +ro22 +2.1 +0.5 +1:I = dgiow +: = +2.5 +JOLJ +2 TEbocJ +0 +500 +400 +Q00 +800 +J000 +0.0 +CE +5.0 +nigisM +04 +a.0 +8.0 +0.IEbocp +0 +52 +20 +J00 +cr1 +500 +0.0 +O'S +04 +nigisM +0.0 +8.0 +CE +0.112 +Xiaoling Zhou et al. +Epoch1 +Epoch10 +Epoch20 +Epoch1 +Epoch30 +Epoch50 +Epoch80 +Epoch100 +Epoch1 +Epoch10 +Epoch20 +Epoch30 +Epoch50 +Epoch80 +Epoch100 +Fig. 6. Top: Equal weights of the two categories. Bottom: Samples in the small category are +assigned with high weights, obtaining better performance for the small (red) category. The im- +balance ratio is set to 10:1. The same conclusions can also be obtained for other imbalance ratios. +Let θλ(w) = θλ(w)/∥θλ(w)∥. Then, it holds that (1) Denote the normalized margin +as +γλ(w)=min +i (fyi(θλ (w) , xi)−max +j̸=i fyj(θλ (w) , xi)) +(10) +Then, γλ (w)→γ∗, as λ → 0. +(2) There exists a λ := λ (r, a, γ∗, w). For α≤2, let θ′(w) denote a α-approximate +minimizer of Lλ. Thus, Lλ +� +θ′ (w) +� +≤ αLλ (θλ (w)). Denote the normalized margin +of θ′(w) by γ′ (w). Then,γ′ (w) ≥ +γ∗ +10αa/r . +The proof is presented in the supplementary file. When λ is sufficiently small, the +difficulty-based weighting does not affect the asymptotic margin. According to Theo- +rem 2, the weights do affect the convergence speed. A good property is that even though +Lλ (θλ (w)) has not yet converged but close enough to its optimum, the corresponding +normalized margin has a reasonable lower bound. A good set of weights can help the +deep learning model to achieve this property faster. However, the conditions in which a +set of weights can accelerate the speed are not clearly illuminated. Notably, as shown in +our experiments in Figs. 4 and A-3 in the supplementary file, assigning large weights for +hard samples increases the convergence speed. The results on the multi-class cases (CI- +FAR10) indicate that assigning large weights on hard samples increases the margin, as +shown in Figs. 5(a-c). However, some particular occasions of difficulty-based weights, +such as SPL [3], do not satisfy the bounding condition because the lower bounds of +these weights are zero instead of a positive real number. The theorem requires further +revision to accommodate this situation. +4.2 +Effects on Generalization Performance +Besides the role of difficulty-based weights on optimization dynamics, we are also con- +cerned as to whether and how the difficulty-based weights affect the generalization +performance. The generalization bound of Xu et al. [20] does not contain the sample +weights, thus it cannot explicitly explain why hard samples are assigned with large +weights. In addition, they assume that the source and target distributions are unequal, +restricting the application of their conclusion. The two generalization bounds we pro- +pose offer good solutions to these issues. They illuminate how a weighting strategies +can be designed. + +Understanding Difficulty-based Sample Weighting with a Universal Difficulty Measure +13 +Let Ps and Pt be the source (training) and target (testing) distributions, respectively, +with the corresponding densities of ps(·) and pt(·). Assume that the two distributions +have the same support. The training and test samples are drawn i.i.d according to dis- +tributions Ps and Pt, respectively. Learning with sample weights w(x) is equivalent +to learning with a new training distribution �Ps. The density of the distribution of the +weighted training set �Ps is denoted as �ps(x) and �ps(x) ∼ w(x)ps(x). Pearson χ2- +divergence is used to measure the difference between �Ps and Pt, i.e., Dχ2(Pt∥ �Ps) = +� +[(d �Ps/dPt)2−1]d �Ps. We consider depth-q (q ≥ 2) networks with the activation func- +tion φ. The binary setting is considered, in that the network computes a real value +f (x) := W qφ (W q−1φ (· · · φ (W 1x) · · · )) , +(11) +where φ(·) is the element-wise activation function (e.g., ReLU). The training set con- +tains n samples. Denote the generalization error for a network f as ˆL(f). The general- +ization performance of f with weights can be described as follows. +Theorem 2. Suppose φ is 1-Lipschitz and 1-positive-homogeneous. With a probability +at least of 1 − δ, we have +ˆL (f) ≤ 1 +n +n +� +i=1 +pt(xi) +�ps(xi)1(yif(xi) < γ) +� +�� +� +I ++ +L · +� +Dχ2 +� +Pt∥ �Ps +� ++ 1 +γ · q(q−1)/2√n +� +�� +� +(II) ++ ϵ(γ, n, δ) +� +�� +� +(III) +, +(12) +where ϵ(γ, n, δ) = +� +log log2 +4L +γ +n ++ +� +log(1/δ) +n +and L:=supx ∥x∥. +The proof is presented in the supplementary file. Compared with the findings of Xu et +al. [20], the bound of the generalization error is directly related to the sample weights +w(x) contained in �ps(x). In view of reducing the generalization error, a natural opti- +mization strategy can be implemented as follows: 1) an optimal weight set w(x) (in +�ps(x)) is obtained according to decreasing the right side of Eq. (12) based on the cur- +rent f; 2) f is then optimized under the new optimal weights w(x). In the first step, +the reduction of generalization error can come from two aspects. One is to increase +the weights of samples with small margins. The other is to make the test and training +distributions close. Disappointingly, this strategy heavily relies on the current f which +is unstable. Given a fixed training set, f depends on random variables (denoted as V) +such as hyperparameters and initialization. To obtain a more stable weighting strategy, +we further propose the following proposition. +Proposition 6. Suppose φ is 1-Lipschitz and 1-positive-homogeneous. With a proba- +bility of at least 1 − δ, we have +EV[ ˆL (fV)] ≤ 1 +n +n +� +i=1 +pt(xi) +�ps(xi)EV[1(yifV(xi) < γ)] +� +�� +� +(I) ++ +L · +� +Dχ2 +� +Pt∥ �Ps +� ++ 1 +γ · q(q−1)/2√n +� +�� +� +(II) ++(III) +(13) + +14 +Xiaoling Zhou et al. +Accordingly, increasing the �ps(xi) of the samples with large EV[1(yifV(xi) < γ)] +will reduce (I). In fact, samples with larger generalization errors will have larger values +of EV[1(yifV(xi) < γ)]. The proof is placed in the supplementary file. Alternatively, +increasing the weights of the hard samples will reduce (I). However, the weights of the +hard samples cannot be increased arbitrarily as Dχ2(Pt∥ �Ps) may be large. Therefore, a +tradeoff between (I) and (II) should be attained to obtain a good set of weights. Alterna- +tively, a good set of weights should increase the weights of hard samples while ensuring +that the distributions of the training set and the test set are close. +It is worth mentioning that our two above conclusions are still insightful when Pt = +Ps while the conclusion of Xu et al. [20] assumes Pt ̸= Ps. Apparently, even when +Pt =Ps, assigning weights according to the samples’ difficulties is still beneficial as the +tradeoff between (I) and (II) still takes effect. +5 +Discussion +Our theoretical analyses in Sections 3 and 4 provide answers to the two concerns de- +scribed in Section 1. +First, the generalization error has been theoretically guaranteed as a generic diffi- +culty measure. It is highly related to noise level, imbalance degree, margin, and uncer- +tainty. Consequently, two directions are worth further investigating. The first direction +pertains to investigating a more efficient and effective estimation method for the gener- +alization error, enhancing its practicality. This will be our future work. As for the second +direction, numerous existing and new weighting schemes can be improved or proposed +using the generalization error as the difficulty measure. Our theoretical findings sup- +plement or even correct the current understanding. For example, samples with large +margins may also be hard-to-classify in some cases (e.g., with heterogeneous samples +in their neighbors). +Second, the existing conclusions on convergence speed have been extended. For +the linear predictors, the existing conclusion is extended by considering our difficulty +measure, namely, the generalization error. For the nonlinear predictors, the conclusion +is extended into the multi-class cases. Furthermore, the explicit relationship between +the generalization gap and sample weights has been established. Our theorem indicates +that assigning large weights on the hard samples may be more effective even when the +source distribution Ps and target distribution Pt are equal. +Our theoretical findings of the generalization bounds provide better explanations to +existing weighting schemes. For example, if heavy noise exists in the dataset, then the +weights of the noisy samples should be decreased. As noisy samples are absent in the +target distribution (i.e., pt(xi) = 0), the weights of the noisy samples in a data set with +heavy noise should be decreased to better match the source and target distributions. The +experiments on the noisy data are shown in Fig. A-5 in which decreasing the weights +of noisy samples obtain the best performance. In imbalanced learning, samples in small +categories have higher errors on the average. Increasing the weights of the hard samples +will not only accelerate the optimization but also improve the performance on the tail +categories, as shown in Figs. 5(d) and 6. These high-level intuitions justify a number +of difficulty-based weighting methods. Easy-first schemes, such as Superloss [7] and + +Understanding Difficulty-based Sample Weighting with a Universal Difficulty Measure +15 +Truncated loss [6], perform well on noisy data. Hard-first schemes, such as G-RW [12] +and Focal Loss [4], are more suitable for imbalanced data. +6 +Conclusion +This study theoretically investigates difficulty-based sample weighting. First, the gen- +eralization error is verified as a universal measure as a means of reflecting the four main +factors influencing the learning difficulty of samples. Second, based on a universal dif- +ficulty measure, the role of the difficulty-based weighting strategy for deep learning is +characterized in terms of convergence dynamics and the generalization bound. Theoret- +ical findings are also presented. Increasing the weights of the hard samples may accel- +erate the optimization. A good set of weights should balance the tradeoff between the +assigning of large weights on the hard samples (heavy training noises are absent) and +keeping the test and the weighted training distributions close. These aspects enlighten +the understanding and design of existing and future weighting schemes. +References +1. Zhou, X., Wu, O.: Which Samples Should be Learned First: Easy or Hard?. arXiv preprint +arXiv:2110.05481 (2021) +2. Khan, S.-H., Hayat, M., Bennamoun, M., Sohel, F.-A., Togneri, R.: Cost-sensitive learning of +deep feature representations from imbalanced data. IEEE Transactions on Neural Networks +and Learning Systems 29(8), 3573–3587 (2018) +3. Kuma, M.-P., Packer, B., Koller, D.: Self-paced learning for latent variable models. In: +NeurIPS, pp. 1–9 (2010) +4. Lin, T.-Y., Goyal, P., Girshick, R., He, K., Dollar, P.: Focal Loss for Dense Object Detection. +IEEE Transactions on Pattern Analysis and Machine Intelligence 42(2), 318–327 (2020) +5. Bengio, Y., Louradour, J., et al.: Curriculum learning. In: ICML, pp. 41–48 (2009) +6. Wang, W., Feng, F., He, X., Nie, L., Chua, T.-S.: Denoising Implicit Feedback for Recom- +mendation. In: WSDM, pp. 373–381 (2021) +7. Castells, T., Weinzaepfel, P., Revaud, J.: SuperLoss: A generic loss for robust curriculum +learning. In: NeurIPS, pp. 1–12 (2020) +8. Emanuel B.-B., Tal R., Nadav Z., Asaf N., Itamar F., Matan P., Lihi Z.-M.: Asymmetric Loss +For Multi-Label Classification. arXiv preprint arXiv:2009.14119 (2020) +9. Santiago, C., Barata, C., Sasdelli, M., et al.: LOW: Training deep neural networks by learning +optimal sample weights. Pattern Recognition 110(1), 107585 (2021) +10. Li, B., Liu, Y., Wang, X.: Gradient Harmonized Single-stage Detector. In: AAAI, pp. 8577– +8584 (2019) +11. Cui, Y., Jia, M., Lin, T.-Y., Song, Y., Belongie, S.: Class-Balanced Loss Based on Effective +Number of Samples. In: CVPR, pp. 9260–9269 (2019) +12. Zhang, S., Li, Z., Yan, S., He, X., Sun, J.: Distribution Alignment: A Unified Framework for +Long-tail Visual Recognition. In: CVPR, pp. 2361–2370 (2021) +13. Zhang, J., Zhu, J., Niu, G., Han, B., Sugiyama, M., Kankanhalli, M.: Geometry-aware +Instance-reweighted Adversarial Training. In: ICLR, pp. 1–29 (2021) +14. Aguilar, E., Nagarajan, B., Khatun, R., Bola˜nos, M., Radeva, P.: Uncertainty modeling and +deep learning applied to food image analysis. In: ICBM, pp. 3–16 (2020) +15. Xiao, Y., Wang, W.-Y. Quantifying uncertainties in natural language processing tasks. In: +AAAI, pp. 7322–7329 (2019) + +16 +Xiaoling Zhou et al. +16. Byrd, J., Lipton, Z.-C.: What is the effect of Importance Weighting in Deep Learning?. In: +ICML, pp. 1405–1419 (2019) +17. Soudry, D., Hoffer, E., Nacson, M.-S., Gunasekar, S., Srebro, N.: The implicit bias of gradi- +ent descent on separable data. Journal of Machine Learning Research 19(1), 1–14 (2018) +18. Chizat, L., Bach, F.: Implicit bias of gradient descent for wide two-layer neural networks +trained with the logistic loss. arXiv preprint arXiv:2002.04486 (2020) +19. Lyu, K., Li, J.: Gradient Descent Maximizes the Margin of Homogeneous Neural Networks. +arXiv preprint arXiv:1906.05890 (2019) +20. Xu, D., Ye, Y., Ruan, C.: Understanding the role of importance weighting for deep learning. +In: ICLR, pp. 1–20 (2020) +21. Goodfellow, I., Bengio, Y., Courville, A.: Deep learning (2016) +22. Heskes, T.: Bias/Variance Decompositions for Likelihood-Based Estimators. Neural Com- +putation 10(6), 1425–1433 (1998) +23. Alex, K., Hinton, G.: Learning multiple layers of features from tiny images. Technical report +(2009) +24. He, K., Zhang, X., Ren S., Sun, J.: Deep Residual Learning for Image Recognition. In: +CVPR, pp. 770–778 (2016) +25. Shu, J., Xie, Q., Yi, L., Zhao, Q., Zhou, S., Xu, Z., Meng, D.: Meta-weight-net: Learning an +explicit mapping for sample weighting. In: NeurIPS, pp. 1–23 (2019) +26. Yang, Z., Yu, Y., You, C., Jacob, S., Yi, M.: Rethinking bias-variance trade-off for general- +ization of neural networks. In: ICML, pp. 10767–10777 (2020) +27. Shin, W., Ha, J.-W., Li S., Cho, Y., et al.: Which Strategies Matter for Noisy Label Classifi- +cation? Insight into Loss and Uncertainty. arXiv preprint arXiv:2008.06218 (2020) +28. Chang, H.-S., Erik, L.-M., McCallum A.: Active bias: Training more accurate neural net- +works by emphasizing high variance samples. In: NeurIPS, pp. 1003–1013 (2017) +29. Swayamdipta, S., Schwartz, R., Lourie, N., Wang, Y., Hajishirzi, H., Smith, N.-A., Choi, Y.: +Dataset cartography: Mapping and diagnosing datasets with training dynamics. arXiv preprint +arXiv:2009.10795 (2020) +30. Agarwal, C., Hooker, S.: Estimating example difficulty using variance of gradients. arXiv +preprint arXiv:2008.11600 (2020) +31. Wolterink, J.-M., Leiner, T., et al.: Generative Adversarial Networks for Noise Reduction in +Low-Dose CT. IEEE Transactions on Medical Imaging 36(12), 2536–2545 (2017) +32. Lowd, D., Meek, C.: Adversarial learning. In: SIGKDD, pp. 641–647 (2005) +33. Elsayed, G.-F., Krishnan, D., Mobahi, H., Regan, K., Bengio, S.: Large margin deep net- +works for classification. In: NeurIPS, pp. 850–860 (2018) +34. Ghasemi, A., Zahediasl, S.: Normality tests for statistical analysis: a guide for non- +statisticians. International journal of endocrinology and metabolism 10(2), 486–489 (2012) +35. Gal, Y., Ghahramani, Z.: Dropout as a bayesian approximation: Representing model uncer- +tainty in deep learning. In: ICML, pp. 1050–1059 (2016) +36. Abdar, M., Pourpanah, F., Hussain, S., Rezazadegan, D., Liu, L., Ghavamzadeh, M., Fieguth, +P., Cao, X., Khosravi, A., Acharya, U.-R., Makarenkov, V., Nahavandi, S.: A review of uncer- +tainty quantification in deep learning: Techniques, applications and challenges. Information +Fusion 76(1), 243–297 (2021) +37. Kendall, A., Gal, Y.: What Uncertainties Do We Need in Bayesian Deep Learning for Com- +puter Vision?. In: NeurIPS, pp. 5575–5585 (2017) + diff --git a/5NE4T4oBgHgl3EQfBQty/content/tmp_files/load_file.txt b/5NE4T4oBgHgl3EQfBQty/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..43d73373a65893894d57fdc502cdfc69a4f01a8b --- /dev/null +++ b/5NE4T4oBgHgl3EQfBQty/content/tmp_files/load_file.txt @@ -0,0 +1,943 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf,len=942 +page_content='Understanding Difficulty-based Sample Weighting with a Universal Difficulty Measure⋆ Xiaoling Zhou1, Ou Wu�1, Weiyao Zhu1, and Ziyang Liang1 Center for Applied Mathematics, Tianjin University, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' {xiaolingzhou,wuou}@tju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='cn, weiyaozhu042@outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='com, ziyangliang@tju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='cn Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Sample weighting is widely used in deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' A large number of weighting methods essentially utilize the learning difficulty of training sam- ples to calculate their weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In this study, this scheme is called difficulty-based weighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Two important issues arise when explaining this scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' First, a uni- fied difficulty measure that can be theoretically guaranteed for training samples does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The learning difficulties of the samples are determined by multiple factors including noise level, imbalance degree, margin, and uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Never- theless, existing measures only consider a single factor or in part, but not in their entirety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Second, a comprehensive theoretical explanation is lacking with respect to demonstrating why difficulty-based weighting schemes are effective in deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In this study, we theoretically prove that the generalization error of a sample can be used as a universal difficulty measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Furthermore, we provide formal theoretical justifications on the role of difficulty-based weighting for deep learning, consequently revealing its positive influences on both the optimization dynamics and generalization performance of deep models, which is instructive to existing weighting schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Keywords: Learning difficulty · Generalization error · Sample weighting · Deep learning interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 1 Introduction Treating each training sample unequally improves the learning performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Two cues are typically considered in designing the weighting schemes of training samples [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The first cue is the application context of learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In applications such as medical diagnosis, samples with high gains/costs are assigned with high weights [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The second cue is the characteristics of the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' For example, samples with low-confidence or noisy labels are assigned with low weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Characteristic-aware weighting has at- tracted increasing attention owing to its effectiveness and universality [3,4,5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Many existing characteristic-aware weighting methods are based on an intrinsic property of the training samples, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', their learning difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The measures for the samples’ learning difficulty can be roughly divided into five categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' ⋆ This study is supported by NSFC 62076178, TJF 19ZXAZNGX00050, and Zhijiang Fund 2019KB0AB03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Paper published at ECML PKDD 2022 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='04850v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='LG] 12 Jan 2023 2 Xiaoling Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' – Prediction-based measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' This category directly uses the loss [3,6,7] or the pre- dicted probability of the ground truth [4,8] as the difficulty measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' This measure is simple yet effective and is widely used in various studies [3,4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Their intention is that a large loss (a small probability) indicates a large learning difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' – Gradient-based measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' This category applies the loss gradient in the measure- ment of the samples’ learning difficulty [9,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Santiagoa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' [9] uses the norm of the loss gradient as the difficulty measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Their intuition is that the larger the norm of the gradient, the harder the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' – Category proportion-based measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' This category is mainly utilized in imbal- anced learning [11], where the category proportion measures the samples’ diffi- culty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' People believe that the smaller the proportion of a category, the larger the learning difficulty of samples in this category [11,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' – Margin-based measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The term “margin” refers to the distance from the sample to the oracle classification boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The motivation is that the smaller the margin, the larger the difficulty of a sample [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' – Uncertainty-based measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' This category uses the uncertainty of a sample to mea- sure the difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Aguilar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' [14] identify hard samples based on epistemic un- certainty and leverage the Bayesian Neural Network [15] to infer it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Varying difficulty measures have a huge impact on a difficulty-based weighting strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The underlying factors which influence samples’ learning difficulty considered in the above measures include noise level [6,7], imbalance degree [11,12], margin [13], and uncertainty [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' However, each measure only considers a single factor or in part, and comes from heuristic inspirations but not formal certifications, hindering the appli- cation scope of the measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' It is desirable to theoretically explore a universal measure capturing all of the above factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Based on this measure, the role of difficulty-based sample weighting can be revealed more concretely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' However, neither theoretical nor empirical investigations have been conducted to investigate a unified measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Moreover, despite the empirical success of various difficulty-based weighting meth- ods, the process of how difficulty-based weighting positively influences the deep learn- ing models remains unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Two recent studies have attempted to investigate the influ- ence of weights in deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Byrd and Lipton [16] empirically studied the train- ing of over-parameterized networks with sample weights and found that these sample weights affect deep learning by influencing the implicit bias of gradient descent-a novel topic in deep learning interpretability, focusing on why over-parameterized models is biased toward solutions that generalize well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Existing studies on this topic [17,18,19] reveal that the direction of the parameters (for linear predictor) and the normalized mar- gin (for nonlinear predictor) respectively converge to those of a max-margin solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Inspired by the finding of Byrd and Lipton [16], Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' [20] dedicated to studying how the understandings for the implicit bias of gradient descent adjust to the weighted empirical risk minimization (ERM) setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' They concluded that assigning high weights to samples with small margins may accelerate optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In addition, they estab- lished a generalization bound for models that implement learning by using sample weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' However, they only discussed the measurement of difficulty by using one of the indicators (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', margin), resulting in that their conclusion is limited and inaccurate in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Furthermore, their generalization bound cannot explicitly explain why Understanding Difficulty-based Sample Weighting with a Universal Difficulty Measure 3 hard samples are assigned with large weights in many studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' More analyses based on a universal difficulty measure are in urgent demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In this study, the manner of how the difficulty-based weighting affects the deep model training is deeply investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' First, our analyses support that the generalization error of the training sample can be regarded as a universal difficulty measure for captur- ing all of the four factors described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Second, based on this unified measure, we characterize the role of difficulty-based weighting on the implicit bias of gradient de- scent, especially for the convergence speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Third, two new generalization bounds are constructed to demonstrate the explicit relationship between the sample weights and the generalization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The two bounds illuminate a new explanation for existing weighting strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Our study takes the first step of constructing a formal theory for difficulty-based sample weighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In summary, our contributions are threefold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' – We theoretically prove the high relevance of the generalization error with four main factors influencing the samples’ learning difficulty, further indicating that the gen- eralization error can be used as a universal difficulty measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' – We reveal how the difficulty-based sample weighting influences the optimization dynamics and the generalization performance for deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Our results indi- cate that assigning high weights on hard samples can not only accelerate the con- vergence speed but also enhance the generalization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' – We bring to light the characteristics of a good set of weights from multiple perspec- tives to illuminate the deep understanding of numerous weighting strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 2 Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='1 Description of Symbols Let X denote the input space and Y a set of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' We assume that the training and test samples are drawn i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='d according to some distributions Dtr and Dte over X × Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The training set is denoted as T = {x, y} = {(xi, yi)}n i=1 that contains n training samples, where xi denotes the i-th sample’s feature, and yi is the associated label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Let di and w (di) be the learning difficulty and the difficulty-based weight of xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The learning difficulty can be approximated by several values, such as loss, uncertainty and generalization error which will be explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The predictor is denoted by f (θ, x) and F = {f (θ, ·) |θ ∈ Θ ⊂ Rd}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' For the sake of notation, we focus on the binary setting yi ∈ {−1, 1} with f (θ, x) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The sign of the model’s output f (θ, xi) is the predicted label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' However, as to be clarified later, our results can be easily extended to the multi-class setting where yi ∈ {1, 2, · · · , C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' For multi-class setting, the softmax function is used to get the probability, and the log- its are given by {fyj (θ, x)}C j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Given a non-negative loss ℓ and a classifier f (θ, ·), the empirical risk can be expressed as L(θ, w) = 1 n �n i=1 w (di) · ℓ (yif (θ, xi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' We focus particularly on the exponential loss ℓ (u) = exp (−u) and logistic loss ℓ (u) = log (1 + exp (−u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Let ∇l(u) be the loss gradient and f (x|T) is the trained model on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The margin is denoted as γi(T) = yif (θ, xi|T) for the binary setting, where it is equivalently denoted as γi(T) = fyi (θ, xi|T) − maxi̸=j fyj (θ, xi|T) for the multi- class setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 4 Xiaoling Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='2 Definition of the Generalization Error Bias-variance tradeoff is a basic theory for the qualitative analysis of the generalization error [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' This tradeoff is initially constructed via regression and mean square error, which is given by Err = Ex,yET [||y − f(x|T)||2 2] ≈ Ex,y[||y − f(x)||2 2] � �� � Bias + Ex,yET [||f(x|T) − f(x)||2 2] � �� � V ariance , (1) where f (x) = ET [f (x|T)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Similarly, we define the generalization error of a single sample xi as erri = ET [ℓ (f (xi|T) , yi)] ≈ B (xi) + V (xi) , (2) where B (xi) and V (xi) are the bias and variance of xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='3 Conditions and Definitions Our theoretical analyses rely on the implicit bias of gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The gradient de- scent process is denoted as θt+1 (w) = θt (w) − ηt∇L (θt [w(d [t])]) , (3) where ηt is the learning rate which can be a constant or step-independent, ∇L (θt [w(d [t])]) is the gradient of L, and w(d [t]) is the difficulty-based weight of difficulty d at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The weight may be dynamic with respect to time t if difficulty measures, such as loss [3] and predicted probability [4], are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' To guarantee the convergence of the gradient descent, two conditions following the most recent study [20] are shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' – The loss ℓ has an exponential tail whose definition is shown in the supplementary file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Thus, limu→∞ ℓ(−u) = limu→∞ ∇ℓ(−u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' – The predictor f(θ, x) is α-homogeneous such that f(c·θ, x) = cαf(θ, x), ∀c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' It is easy to verify that losses including the exponential loss, log loss, and cross-entropy loss satisfy the first condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The second condition implies that the activation functions are homogeneous such as ReLU and LeakyReLU, and bias terms are disallowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In addition, we need certain regularities from f(θ, x) to ensure the existence of critical points and the convergence of gradient descent: – For ∀x∈X, f(θ, x) is β-smooth and l-Lipschitz on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The third condition is a common technical assumption whose practical implications are discussed in the supplementary file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The generalization performance of deep learning models is measured by the gener- alization error of the test set ˆL (f) [21], defined as ˆL (f) = P(x,y)∼Dte[γ(f (x, y)) ≤ 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' (4) Understanding Difficulty-based Sample Weighting with a Universal Difficulty Measure 5 0 50 100 150 200 0 1 2 3 4 5 Error Id 0 100 200 300 400 2 0 2 4 6 8 10 12 14 16 18 Error Id Noise Clean 1 (Largest) 2 3 4 5 6 7 8 9 10 (Smallest) 1 (Largest) 2 3 4 5 6 7 8 9 10 (Smallest) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' (a) Generalization errors of clean and noisy samples on noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The noise ratio is 10% (b) Generalization errors of samples in ten categories on imbalanced data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The imbalance ratio is 10:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' CIFAR10 and ResNet32 are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Other values of noise ratio and imbalance ratio following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' [25] are also experimented with and the same conclusions can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='4 Experiment Setup Demonstrated experiments are performed to support our theoretical analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' For the simulated data, the linear predictor is a regular regression model, and the nonlinear pre- dictor is a two-layer MLP with five hidden units and ReLU as the activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Exponential loss and standard normal initialization are utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' CIFAR10 [23] is exper- imented with, and ResNet32 [24] is adopted as the baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' For the imbalanced data, the imbalance setting follows Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' For the noisy data, uniform and flip label noises are used and the noise setting follows Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The models are trained with a gradient descent by using 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='1 as the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The model uncertainty is approximated by the predictive variance of five predic- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' To approximate the generalization error, we adopt the five-fold cross-validation [26] to calculate the average learning error for each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 3 A Universal Difficulty Measure As previously stated, four factors pointed out by existing studies, namely, noise, imbal- ance, margin, and uncertainty, greatly impact the learning difficulty of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Nev- ertheless, existing measures only consider one or part of them, and their conclusions are based on heuristic inspirations and empirical observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In this section, we theo- retically prove that the generalization error of samples is a universal difficulty measure reflecting all four factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' All proofs are presented in the supplementary file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Without increasing the ambiguity, the generalization error of the samples is termed as error for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='1 Noise Factor Noise refers to data that is inaccurate in describing the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Numerous studies devoted to reducing the influence of noisy samples in the dataset on the deep learning models 6 Xiaoling Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' and these literature intuitively consider noisy samples as hard ones without formal cer- tification [7,27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The two kinds of noise are feature noise [31] and label noise [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' We offer two propositions to reveal the relationship between the generalization error and the noise factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' For feature noise, we offer the following proposition: Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Let ∆xi be the perturbation of sample (xi, yi), which is extremely small in that o(∆xi) can be omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Let ∠ϕ be the angle between the direction of ∆xi and the direction of ET [f ′ (xi|T)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' If ET [f ′ (xi|T) · ∆xi] < 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', ∠ϕ > 90◦), then the error of the noisy sample is increased relative to the clean one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Alternatively, the direction of the perturbation ∆xi and that of ET [f ′ (xi|T)] are contradictory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Oth- erwise, if ET [f ′ (xi|T) · ∆xi] > 0, then ∠ϕ < 90◦, and the error of the noisy sample is decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' According to Proposition 1, feature noise can be divided into two categories, which increase or decrease the learning difficulty (generalization error) of the samples, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In this paper, noise that increases the error is called the adversarial type, which is always used in the field of adversarial learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' otherwise, it is a promoted type, which refers to noise that decrease the learning difficulty of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Therefore, the variation of the error under feature noise is determined by the noise type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' For example, as all feature noises are adversarial in adversarial learning [32], all of the samples’ errors are increased with feature noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' For label noise, we offer the following proposition: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Let π be the label corruption rate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', the probability of each label flipping to another one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Denote the probability of correct classification for the original samples as p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' If p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='5, then the errors of the noisy samples are larger than those of the clean ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' This proposition implies that the errors of the samples with label noises are larger than those of the clean ones on the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Specifically, if a sample is more likely to be predicted correctly, its generalization error is increased due to label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Let L∗ be the global optimum of the generalization error of the clean dataset and y′ be the corrupted label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' When the noise in Proposition 2 is added, the empirical error L′ is L′ = (1 − π) L∗ + πL (f (x) , y′) , (5) where we have taken expectations over the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' When π → 0, the noise disappears, and the optimal generalization is attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Proposition 2 is consistent with the empirical observation shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 1(a), where the noisy samples have larger errors than the clean ones on the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='2 Imbalance Factor Besides noise, imbalance is another common deviation of real world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The cat- egory distribution of the samples in the training set is non-uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Various methods solve this issue by assigning high weights on samples in tail categories which are con- sidered to be hard ones [4,11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Nevertheless, a theoretical justification about why these samples are harder lacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The imbalance ratio is denoted by cr =max{c1, c2, · · · , cC}: min{c1, c2, · · · , cC}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Then, we offer the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Understanding Difficulty-based Sample Weighting with a Universal Difficulty Measure 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' (a) Correlation between generalization error and average margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' (b) Correlation between generalization error and epistemic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' CIFAR10 and ResNet32 are used in this experi- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' All values are normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' If a predictor on an imbalanced dataset (cr > e : 1) is an approximate Bayesian optimal classifier (as the exponential loss is an approximation for the zero- one loss), which is to minimize the total risk, then the average probability of the ground truth of the samples in the large category is greater than that of the samples in the small category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' With Proposition 3, it is easy to obtain Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='1 shown in the supplemen- tary file that the average error of samples in the small category is larger than that of the samples in the large category, indicating there are more hard samples in the small category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' This proposition is verified by the experiments, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The tail categories contain more samples with larger errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' To enhance the performance of the classification model, samples with larger errors should be assigned with higher weights, as most methods do [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Further experiments in Section 5 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 6) indicate that the classification performance of the small category can be improved by increasing its sample weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='3 Margin Factor The samples’ margins measure the distances of the samples from the decision boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Some literature intuitively consider a small margin indicates a large learning difficulty and corresponds to a low confidence of the prediction [33,13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' However, a formal justi- fication is lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' We offer the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Let µi be the true margin of xi corresponding to the oracle decision boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The condition is that the functional margins of a sample trained on random datasets obey a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In other words, for sample xi, its functional margin γi obey a Gaussian distribution N(µi, σ2 i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' For sample xj, γj ∼ N(µj, σ2 j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' rwr!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' www hEULOL(p)WIDIDMA wypV ELLOL igsMELLOLELLOLbI8 Xiaoling Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The distributions of samples’ margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' when the margin variances of the two samples are same (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', σ2 i = σ2 j ), if µi ≤ µj, then erri ≥errj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Similarly, when the true margins of the two samples are the same (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', µi =µj), if σ2 i ≥σ2 j , then erri ≥errj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Proposition 5 indicates a fact that even a sample with a large true margin, as long as the margin variance is large, it may also have a high learning difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Specifically, the true margin (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', the mean of the functional margin distribution) of a sample and error are negatively correlated when the margin variances of the samples are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' By contrast, the margin variance and error are positively correlated when the true margins are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' This illumination revises the current wisdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The conclusion in which sam- ples close to the oracle decision boundary are hard ones [20] is not completely correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Indeed, the relation between the margin and error of sample xi conforms with the fol- lowing formula: erri = ET [e−γi(T )] = e−µi+ 1 2 σ2 i , (6) where erri, µi, and σi refer to the generalization error, the true margin, and the margin variance of sample xi, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' For the two samples xi and xj, if µi < µj and σ2 i < σ2 j , then we cannot judge whether erri is greater than errj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 2(a), the average margin and error are negatively correlated for most samples, but it is not absolute, which accords with the above analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Although it is intuitive that the func- tional margin trained on random datasets obeys a Gaussian distribution, we evaluate it via the Z-scores of the distributions’ Kurtosis and Skewness [34] which is shown in Fig 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' More margin distribution curves and all Z-score values of the distributions are shown in the supplementary file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' As all Z-scores are in [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='96, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='96], under the test level of α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='05, the distribution of margin obeys the Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='4 Uncertainty Factor Uncertainties [37] in deep learning are classified into two types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The first type is aleatoric uncertainty (data uncertainty), which is caused by the noise in the observation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Its correlation with the error has been discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The second type is epis- temic uncertainty (model uncertainty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' It is used to indicate the consistency of multiple predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' We give the analyses of the relationship between the generalization error and epistemic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Let T be a training set, and let P(θ|T) be the distribution of the training models based on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=" The predictive variance V ar(f(xi|θ1), · · · , f(xi|θK)) plus a precision '00000000 T0000000 S0000000 30000000 40000000 20000000 0 JO- ELGUdIGUC2 50- 30rtigsM T0000000 S0000000 30000000 40000000 20000000 0 JO- ELGUdGUcA 30- 30T2000000 S0000000 32000000 30000000 32000000 0000000 42000000 0 JO- ELGUdnIGUCA 30- 30- 如-30000000 35000000 34000000 3Q000000 ELGdtGUcA JO- J2- 30- 32-tigsM SS000000 000000ES." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 54000000 32000000 S2000000 S3000000 58000000 0 JO- ELedGUcA 30- 30rtigrsM 55000000 53000000 Q000002S 5Q000000 000000TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 58000000 JO- 50- 30-30000000 35000000 34000000 3Q000000 2- J O- J2- 50- 32-WSa 12000000 50000000 00000025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' QQQQQQ0E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 32000000 Q000000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' JO- 5O 30- 40-rtigisM J0000000 50000000 30000000 40000000 20000000 10- 50- 30- 40-WS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='00000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='J0000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='30000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='30000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='40000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='20000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='JO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='5O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='30rigrsM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='00000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='J0000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='50000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='30000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='40000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='20000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='JO- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='50- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='30- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='40-rtigrsM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='30000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='32000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='30000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='32000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='40000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='42000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='JO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='J2- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='30-rtigisM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='J0000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='50000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='30000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='40000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='20000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='J O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='J2- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='30ntigrsM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='T0000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='S0000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='30000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='40000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='20000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='2- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='ELGUdGUCA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='10- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='J2- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='30-S0000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='$2000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='30000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='32000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='42000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='ELGUdGUc2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='10- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='J2- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='30-tigisM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content="'00000000 " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='T0000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='S0000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='30000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='40000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='20000000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='JO- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='ELGUdGUcA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='30- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='30- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='如-Understanding Difficulty-based Sample Weighting with a Universal Difficulty Measure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='constant is a typical manner of estimating epistemic uncertainty [35,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Take the mean square loss as an example1, the epistemic uncertainty is � Var [xi] :=τ −1 + 1 |K| � k f(xi|θk)⊺f(xi|θk) − E[f(xi|θk)]⊺E[f(xi|θk)], (7) where τ is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The second term on the right side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' (7) is the second raw mo- ment of the predictive distribution and the third term is the square of the first moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' When K → ∞ and the constant term is ignored, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' (7) becomes � Var [xi] := � θ ||f(xi|θ) − f(xi)||2 2dP(θ|T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' (8) If P(θ|T) is approximated by the distribution of learned models on random training sets which conform to the Gaussian distribution N(T, δI), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' (8) is exactly the variance term of the error defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' (2) when the mean square loss is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' As the bias term in the error can capture the aleatoric uncertainty and the variance term captures the epistemic uncertainty, the overall relationship between uncertainty and error is positively correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Nevertheless, the relationship between epistemic un- certainty and error is not simply positively or negatively correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' For some samples with heavy noises, their epistemic uncertainties will be small as their predictions remain erroneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' However, their errors are large due to their large bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' This phenomenon is consistent with the experimental results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Epistemic uncertainty and error are positively correlated for some samples, and the two variables are negatively correlated for other samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='5 Discussion about Generalization Error The commonly used difficulty measures, such as loss [3] and gradient norm [9], are mainly related to the bias term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Shin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' [27] emphasized that only using loss as the measurement cannot distinguish clean and noisy samples, especially for uniform la- bel noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' There are also a few existing studies that use variance [28,29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' For instance, Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' [30] applied the variance of gradient norms as the difficulty measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Indeed, both the variance and bias terms should not be underestimated when measur- ing the samples’ learning difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Our theoretical analyses support that generalization error including both the two terms can capture four main factors influencing the sam- ples’ learning difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Thus, the error can be leveraged as a universal measure that is more reasonable than existing measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Existing studies generally apply the K-fold cross-validation method [26] to calculate the generalization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' More efficient error calculation algorithms are supposed to be proposed which will be our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 4 Role of Difficulty-Based Weighting This section aims to solve the second issue of explaining the difficulty-based weighting in deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Based on the universal difficulty measure, the impacts of the difficulty- based weighting schemes on the optimization dynamics and the generalization perfor- mance in deep learning are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Compared with the most recent conclusions 1 For other losses, other methods can be used to calculate the predictive variance [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 10 Xiaoling Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' “Cosine distance” represents the cosine of the angle between the decision boundary (at that epoch) and the max-margin solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' (a), (b) Cosine distance and average margin of equal weights and inverse margin weights using the linear predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' (c), (d) Cosine distance and average margin of equal weights and inverse margin weights using the nonlinear predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' (e), (f) Cosine distance and average margin of equal weights and increasing weights of noisy samples using the nonlinear predictor on the noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' (g), (h) Cosine distance and average margin of equal weights and increasing weights of samples in tail categories using the linear predictor on the imbalanced data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' More results are placed in the supplementary file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' [20] established only on the margin factor, our theoretical findings, which are based on our universal measure, are more applicable and precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='1 Effects on Optimization Dynamics Linear Predictor We begin with the linear predictors allowing for a more refined analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' [20] inferred an upper bound containing the term DKL(p∥w), where DKL is the Kullback-Leibler divergence and p is the optimal dual coefficient vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' A smaller value of DKL(p∥w) means that the convergence may be accelerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' There- fore, to accelerate the convergence, they believe that the weights w should be consistent with the coefficients p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Alternatively, the samples with small functional margins will have large coefficients and thus should be assigned with large weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' However, the functional margin is not the true margin that corresponds to the oracle boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' There- fore, their conclusion that samples close to the oracle classification boundary should be assigned with large weights [20] cannot be well-drawn according to their inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' We offer a more precise conclusion with the unified difficulty measure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', generalization error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' As before, we assume that the functional margins of a sample xi obey a Gaus- sian distribution N(µi, σ2 i ), where µi is the true margin and σ2 i is the margin variance of xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' We offer the following proposition: Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' For two samples xi and xj, if erri ≥ errj, then we have: (1) When the optimal dual coefficient pi of xi on a random training set T is a linear function of its functional margin γi on T, if µi ≤ µj, then ET [pi] ≥ ET [pj] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', ET [wi] ≥ ET [wj]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=" S(p)Ebocj0'5 - 0°4 - 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content="0 I'O -batdgisw Isupg 0 500 400 00 008 J000 0 500 400 e00 008 J000 028." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content="0 0'S 28." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 0a52 020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 zre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='1 I000 oitogrib Ismitqo ot onistaib 2o0 2nigisM(Ol:) batdgiaw sl baoslsdmi batdgisw Isupg 0 500 400 e00 008 1000 0 500 400 e00 800 J000 F 0e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 F I- E se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 F 0 I Ae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 Foe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 3 F 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 4 F 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content="1 oitogrib Ismitqo ot onstaib 2o0 2nigisMbatdgigw Isupg 0 500 400 e00 800 J000 0 500 400 e00 800 J000 0'2 5." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content="0 0'4 - 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=" 1 I'S 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content="1 2nigisMcbatdgigw Isupe (OI:1) batdgiw 22slo baonslsdmi 0 500 400 eoo 800 000 0 500 400 e00 008 J000 I - 0'4 0 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 I - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 5- 3 - 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 4 - 2 1 noitogrib Ismitqo ot gonstaib 2o0 2nigisM(s) Eboc (p) Ebocj 10 S00 400 00a 008 1000 0 S00 400 e00 800 1000 batdgigw Isupg batdgigw Isupg 08e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content="0 0'4 - 280." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 0Qe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content="0 0002 - I'0 000." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content="1 goib zo 2nigisM(c) Ebocj (g) Ebocj 10 500 400 e00 008 1000 500 400 000 008 1000 0'3 batdgigw Isupg I - badgig Isupg +." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content="0 (OI:1) batdgigw 2esl baonslsdmi (0I: I) batdgigw 2slo baoslsdmi 0 - 0'2 I - 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 5- 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 3 - e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content="0 4 - I'0 - 2 - gosib izo 2nigisM(3) Ebocj () Ebocj :0 S00 400 e00 008 1000 10 S00 400 00a 008 J000 08." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 batdgisw Isupg 0°4 batdgigw ionl 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 0e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 F 2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 1 I"I F 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=" 1 I'S." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' ib io 2nigisM(a) Ebocj (p) Ebocj 0: S00 400 e00 800 1000 0: S00 400 00a 008 J000 2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 batdgigw Isupg batdgigw ionl F 08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 F 2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='1 F 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='1 上 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='1 ib io 2nigisMEbocp(α)((L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='Ebocp Ebocp 0 S00 400 000 800 1000 10 S00 400 e00 800 1000 F0 batdgigw Isupg batdgigw Isupe 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 (1:1) batdgigw 2esl baoslsdmi I (01:1) batdgigw 2eslo baoslsdmi 0e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 5- F se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 3 F Ae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 4 F ae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 2 - F80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content="0 I00 - ostaib nizo 2nigisMEbocp Ebocp 10 500 400 e00 008 1000 10 500 400 e00 008 1000 0'2 2r." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 batdgigw Isupg batdgiow Isup 0°20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content="0 0'52 - 0' 00." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content="0 0'2 8." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 zr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content="0 I'00 - I'0 - oib izo 2nigisMbatdgigw Isupg 0 500 400 e00 800 1000 0 S00 400 e00 800 1000 88." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 + +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 0e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 F se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 F e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='1 0e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content="0 I'S 8e." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content="0 I'4 F 00." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='1 g01stzib 200 2nigisMUnderstanding Difficulty-based Sample Weighting with a Universal Difficulty Measure 11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' (a)-(c) Normalized margin of increasing the weights of noisy samples/samples with small margins/samples in tail categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' CIFAR10 data is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Uniform label noise is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The noise ratio and imbalance ratio are 10% and 10:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' (d) Generalization error of the test set when the nonlinear model is trained with different weights on simulated imbalanced data with the imbalance ratio as 10:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Other noise and imbalance settings are also experimented with and the same conclusions can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' (2) When the optimal dual coefficient pi of xi on a random training set T is a natural exponential function of its functional margin γi on T, ET [pi] ≥ ET [pj] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', ET [wi] ≥ ET [wj]) always holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Notably, even when µi > µj, ET [pi] > ET [pj] may still hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The proof is presented in the supplementary file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' ET [pi] > ET [pj] implies that wi > wj holds on the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The conclusion that samples with small true margins should be assigned with large weights may not hold on some training sets when pi is not a linear function of γi [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' A sample with a small true margin may have a smaller weight than a sample with a large true margin yet a large error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Thus, a more general conclusion when pi is not a linear function of γi is that increasing the weights of hard samples (samples with large generalization errors) may accelerate the convergence, rather than just for samples with small margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Other factors, including noise, imbal- ance, and uncertainty also affect samples’ learning difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Notably, the weights of the hard samples should not be excessively increased, as to be explained in the succeed- ing section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' We reasonably increase the weights of the hard samples shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 4 and A-3 in the supplementary file indicating that the optimization is accelerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' We also prove that difficulty-based weights do not change the convergence direction to the max-margin solution shown in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='1 in the supplementary file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 3, the cosine distance and margin value are always increasing during the training procedure, indicating the direction of the asymptotic margin is the max-margin solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Nonlinear Predictor Analyzing the gradient dynamics of the nonlinear predictors is insurmountable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The main conclusion obtained by Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' [20] can also be established for difficulty-based weights only if the bound of weights is larger than zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' However, their theorem has only been proven for binary cases as the employed loss is inapplicable in multi-class cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Here, we extend the theory to the multi-class setting with a regu- larization λ||θ||r on the cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Let θλ (w)∈arg min Lλ (θ, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Formally, the dynamic regime for the nonlinear predictor can be described as follows: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Let w ∈ [b, B]n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Denote the optimal normalized margin as γ∗ = max ∥θ(w)∥≤1 min i (fyi(θ(w), xi) − max j̸=i (fyj(θ(w), xi))) (9) Ebocp 0 500 400 00 800 J000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 CE 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 04 nigisM a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='1Ebocp 0 52 0 J00 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content="0 o'S 04 migisM a." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 CE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='1 O12G Icieg2u e Meia2 ot Jo12a 2bje2Ebocp2 0 J2 J00 J52 J20 r 500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='5 01:1 = gigw 2:I = dgigw 52 Ismrronl 2 TEbocpEbocp 0 J00 500 300 400 200 e00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content="0 CE 0'4 nigsM a." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='19PCEbocp 0 52 20 J00 cr1 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content="0 O'S 0'4 nigisM 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 CE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='1Ebocp 0 02 J00 J20 S00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 migisM 04 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 M 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 CE V0126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='1Ebocp 0 J00 J20 r 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content="0 0'5 nigisM 04 a." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 CE IpgJSUcG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='1Ebocp 0 5O 40 80 J00 JSO J40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 VOSIUOA 『.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 WM 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0Ebocp 0 40 80 J00 JSO J40 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 0°004 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 2201 800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content="0 O'OJS 0'014Ebocp 0 SO 40 eo 80 J00 JSO J40 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 nigisM 04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='1EbocpEbocp 0 52 02 J00 s J20 r 500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='I ro22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='5 1:I = dgiow : = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='5 JOLJ 2 TEbocJ 0 500 400 Q00 800 J000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 CE 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 nigisM 04 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='IEbocp 0 52 20 J00 cr1 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content="0 O'S 04 nigisM 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='0 CE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='112 Xiaoling Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Epoch1 Epoch10 Epoch20 Epoch1 Epoch30 Epoch50 Epoch80 Epoch100 Epoch1 Epoch10 Epoch20 Epoch30 Epoch50 Epoch80 Epoch100 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Top: Equal weights of the two categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Bottom: Samples in the small category are assigned with high weights, obtaining better performance for the small (red) category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The im- balance ratio is set to 10:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The same conclusions can also be obtained for other imbalance ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Let θλ(w) = θλ(w)/∥θλ(w)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Then, it holds that (1) Denote the normalized margin as γλ(w)=min i (fyi(θλ (w) , xi)−max j̸=i fyj(θλ (w) , xi)) (10) Then, γλ (w)→γ∗, as λ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' (2) There exists a λ := λ (r, a, γ∗, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' For α≤2, let θ′(w) denote a α-approximate minimizer of Lλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Thus, Lλ � θ′ (w) � ≤ αLλ (θλ (w)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Denote the normalized margin of θ′(w) by γ′ (w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Then,γ′ (w) ≥ γ∗ 10αa/r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The proof is presented in the supplementary file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' When λ is sufficiently small, the difficulty-based weighting does not affect the asymptotic margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' According to Theo- rem 2, the weights do affect the convergence speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' A good property is that even though Lλ (θλ (w)) has not yet converged but close enough to its optimum, the corresponding normalized margin has a reasonable lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' A good set of weights can help the deep learning model to achieve this property faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' However, the conditions in which a set of weights can accelerate the speed are not clearly illuminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Notably, as shown in our experiments in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 4 and A-3 in the supplementary file, assigning large weights for hard samples increases the convergence speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The results on the multi-class cases (CI- FAR10) indicate that assigning large weights on hard samples increases the margin, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 5(a-c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' However, some particular occasions of difficulty-based weights, such as SPL [3], do not satisfy the bounding condition because the lower bounds of these weights are zero instead of a positive real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The theorem requires further revision to accommodate this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='2 Effects on Generalization Performance Besides the role of difficulty-based weights on optimization dynamics, we are also con- cerned as to whether and how the difficulty-based weights affect the generalization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The generalization bound of Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' [20] does not contain the sample weights, thus it cannot explicitly explain why hard samples are assigned with large weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In addition, they assume that the source and target distributions are unequal, restricting the application of their conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The two generalization bounds we pro- pose offer good solutions to these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' They illuminate how a weighting strategies can be designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Understanding Difficulty-based Sample Weighting with a Universal Difficulty Measure 13 Let Ps and Pt be the source (training) and target (testing) distributions, respectively, with the corresponding densities of ps(·) and pt(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Assume that the two distributions have the same support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The training and test samples are drawn i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='d according to dis- tributions Ps and Pt, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Learning with sample weights w(x) is equivalent to learning with a new training distribution �Ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The density of the distribution of the weighted training set �Ps is denoted as �ps(x) and �ps(x) ∼ w(x)ps(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Pearson χ2- divergence is used to measure the difference between �Ps and Pt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Dχ2(Pt∥ �Ps) = � [(d �Ps/dPt)2−1]d �Ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' We consider depth-q (q ≥ 2) networks with the activation func- tion φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The binary setting is considered, in that the network computes a real value f (x) := W qφ (W q−1φ (· · · φ (W 1x) · · · )) , (11) where φ(·) is the element-wise activation function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', ReLU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The training set con- tains n samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Denote the generalization error for a network f as ˆL(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The general- ization performance of f with weights can be described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Suppose φ is 1-Lipschitz and 1-positive-homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' With a probability at least of 1 − δ, we have ˆL (f) ≤ 1 n n � i=1 pt(xi) �ps(xi)1(yif(xi) < γ) � �� � I + L · � Dχ2 � Pt∥ �Ps � + 1 γ · q(q−1)/2√n � �� � (II) + ϵ(γ, n, δ) � �� � (III) , (12) where ϵ(γ, n, δ) = � log log2 4L γ n + � log(1/δ) n and L:=supx ∥x∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The proof is presented in the supplementary file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Compared with the findings of Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' [20], the bound of the generalization error is directly related to the sample weights w(x) contained in �ps(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In view of reducing the generalization error, a natural opti- mization strategy can be implemented as follows: 1) an optimal weight set w(x) (in �ps(x)) is obtained according to decreasing the right side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' (12) based on the cur- rent f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 2) f is then optimized under the new optimal weights w(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In the first step, the reduction of generalization error can come from two aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' One is to increase the weights of samples with small margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The other is to make the test and training distributions close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Disappointingly, this strategy heavily relies on the current f which is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Given a fixed training set, f depends on random variables (denoted as V) such as hyperparameters and initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' To obtain a more stable weighting strategy, we further propose the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Suppose φ is 1-Lipschitz and 1-positive-homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' With a proba- bility of at least 1 − δ, we have EV[ ˆL (fV)] ≤ 1 n n � i=1 pt(xi) �ps(xi)EV[1(yifV(xi) < γ)] � �� � (I) + L · � Dχ2 � Pt∥ �Ps � + 1 γ · q(q−1)/2√n � �� � (II) +(III) (13) 14 Xiaoling Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Accordingly, increasing the �ps(xi) of the samples with large EV[1(yifV(xi) < γ)] will reduce (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In fact, samples with larger generalization errors will have larger values of EV[1(yifV(xi) < γ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The proof is placed in the supplementary file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Alternatively, increasing the weights of the hard samples will reduce (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' However, the weights of the hard samples cannot be increased arbitrarily as Dχ2(Pt∥ �Ps) may be large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Therefore, a tradeoff between (I) and (II) should be attained to obtain a good set of weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Alterna- tively, a good set of weights should increase the weights of hard samples while ensuring that the distributions of the training set and the test set are close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' It is worth mentioning that our two above conclusions are still insightful when Pt = Ps while the conclusion of Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' [20] assumes Pt ̸= Ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Apparently, even when Pt =Ps, assigning weights according to the samples’ difficulties is still beneficial as the tradeoff between (I) and (II) still takes effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 5 Discussion Our theoretical analyses in Sections 3 and 4 provide answers to the two concerns de- scribed in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' First, the generalization error has been theoretically guaranteed as a generic diffi- culty measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' It is highly related to noise level, imbalance degree, margin, and uncer- tainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Consequently, two directions are worth further investigating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The first direction pertains to investigating a more efficient and effective estimation method for the gener- alization error, enhancing its practicality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' This will be our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' As for the second direction, numerous existing and new weighting schemes can be improved or proposed using the generalization error as the difficulty measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Our theoretical findings sup- plement or even correct the current understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' For example, samples with large margins may also be hard-to-classify in some cases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', with heterogeneous samples in their neighbors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Second, the existing conclusions on convergence speed have been extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' For the linear predictors, the existing conclusion is extended by considering our difficulty measure, namely, the generalization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' For the nonlinear predictors, the conclusion is extended into the multi-class cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Furthermore, the explicit relationship between the generalization gap and sample weights has been established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Our theorem indicates that assigning large weights on the hard samples may be more effective even when the source distribution Ps and target distribution Pt are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Our theoretical findings of the generalization bounds provide better explanations to existing weighting schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' For example, if heavy noise exists in the dataset, then the weights of the noisy samples should be decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' As noisy samples are absent in the target distribution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', pt(xi) = 0), the weights of the noisy samples in a data set with heavy noise should be decreased to better match the source and target distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' The experiments on the noisy data are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' A-5 in which decreasing the weights of noisy samples obtain the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In imbalanced learning, samples in small categories have higher errors on the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Increasing the weights of the hard samples will not only accelerate the optimization but also improve the performance on the tail categories, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 5(d) and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' These high-level intuitions justify a number of difficulty-based weighting methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Easy-first schemes, such as Superloss [7] and Understanding Difficulty-based Sample Weighting with a Universal Difficulty Measure 15 Truncated loss [6], perform well on noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Hard-first schemes, such as G-RW [12] and Focal Loss [4], are more suitable for imbalanced data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 6 Conclusion This study theoretically investigates difficulty-based sample weighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' First, the gen- eralization error is verified as a universal measure as a means of reflecting the four main factors influencing the learning difficulty of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Second, based on a universal dif- ficulty measure, the role of the difficulty-based weighting strategy for deep learning is characterized in terms of convergence dynamics and the generalization bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Theoret- ical findings are also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Increasing the weights of the hard samples may accel- erate the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' A good set of weights should balance the tradeoff between the assigning of large weights on the hard samples (heavy training noises are absent) and keeping the test and the weighted training distributions close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' These aspects enlighten the understanding and design of existing and future weighting schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Zhou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Wu, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': Which Samples Should be Learned First: Easy or Hard?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='. arXiv preprint arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='05481 (2021) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Khan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Hayat, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Bennamoun, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Sohel, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Togneri, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': Cost-sensitive learning of deep feature representations from imbalanced data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' IEEE Transactions on Neural Networks and Learning Systems 29(8), 3573–3587 (2018) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Kuma, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Packer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Koller, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': Self-paced learning for latent variable models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In: NeurIPS, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 1–9 (2010) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Lin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Goyal, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Girshick, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', He, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Dollar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': Focal Loss for Dense Object Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' IEEE Transactions on Pattern Analysis and Machine Intelligence 42(2), 318–327 (2020) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Bengio, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Louradour, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' : Curriculum learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In: ICML, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 41–48 (2009) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Feng, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', He, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Nie, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Chua, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': Denoising Implicit Feedback for Recom- mendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In: WSDM, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 373–381 (2021) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Castells, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Weinzaepfel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Revaud, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': SuperLoss: A generic loss for robust curriculum learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In: NeurIPS, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 1–12 (2020) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Emanuel B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Tal R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Nadav Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Asaf N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Itamar F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Matan P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Lihi Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': Asymmetric Loss For Multi-Label Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' arXiv preprint arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='14119 (2020) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Santiago, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Barata, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Sasdelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' : LOW: Training deep neural networks by learning optimal sample weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Pattern Recognition 110(1), 107585 (2021) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Li, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': Gradient Harmonized Single-stage Detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In: AAAI, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 8577– 8584 (2019) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Cui, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Jia, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Lin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Song, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Belongie, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': Class-Balanced Loss Based on Effective Number of Samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In: CVPR, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 9260–9269 (2019) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Yan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', He, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Sun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': Distribution Alignment: A Unified Framework for Long-tail Visual Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In: CVPR, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 2361–2370 (2021) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Zhu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Niu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Han, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Sugiyama, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Kankanhalli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': Geometry-aware Instance-reweighted Adversarial Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In: ICLR, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 1–29 (2021) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Aguilar, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Nagarajan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Khatun, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Bola˜nos, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Radeva, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': Uncertainty modeling and deep learning applied to food image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In: ICBM, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 3–16 (2020) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Xiao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Quantifying uncertainties in natural language processing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In: AAAI, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 7322–7329 (2019) 16 Xiaoling Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Byrd, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Lipton, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': What is the effect of Importance Weighting in Deep Learning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='. In: ICML, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 1405–1419 (2019) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Soudry, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Hoffer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Nacson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Gunasekar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Srebro, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': The implicit bias of gradi- ent descent on separable data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Journal of Machine Learning Research 19(1), 1–14 (2018) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Chizat, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Bach, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': Implicit bias of gradient descent for wide two-layer neural networks trained with the logistic loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' arXiv preprint arXiv:2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='04486 (2020) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Lyu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': Gradient Descent Maximizes the Margin of Homogeneous Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' arXiv preprint arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='05890 (2019) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Xu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Ye, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Ruan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': Understanding the role of importance weighting for deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In: ICLR, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 1–20 (2020) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Goodfellow, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Bengio, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Courville, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': Deep learning (2016) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Heskes, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': Bias/Variance Decompositions for Likelihood-Based Estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Neural Com- putation 10(6), 1425–1433 (1998) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Alex, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Hinton, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': Learning multiple layers of features from tiny images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Technical report (2009) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' He, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Ren S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Sun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': Deep Residual Learning for Image Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In: CVPR, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 770–778 (2016) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Shu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Xie, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Yi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Zhao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Zhou, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Xu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Meng, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': Meta-weight-net: Learning an explicit mapping for sample weighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In: NeurIPS, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 1–23 (2019) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Yu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', You, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Jacob, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Yi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': Rethinking bias-variance trade-off for general- ization of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In: ICML, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 10767–10777 (2020) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Shin, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Ha, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Li S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Cho, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' : Which Strategies Matter for Noisy Label Classifi- cation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Insight into Loss and Uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' arXiv preprint arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='06218 (2020) 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Chang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Erik, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', McCallum A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': Active bias: Training more accurate neural net- works by emphasizing high variance samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In: NeurIPS, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 1003–1013 (2017) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Swayamdipta, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Schwartz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Lourie, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Hajishirzi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Smith, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Choi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': Dataset cartography: Mapping and diagnosing datasets with training dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' arXiv preprint arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='10795 (2020) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Agarwal, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Hooker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': Estimating example difficulty using variance of gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' arXiv preprint arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='11600 (2020) 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Wolterink, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Leiner, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' : Generative Adversarial Networks for Noise Reduction in Low-Dose CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' IEEE Transactions on Medical Imaging 36(12), 2536–2545 (2017) 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Lowd, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Meek, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': Adversarial learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In: SIGKDD, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 641–647 (2005) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Elsayed, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Krishnan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Mobahi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Regan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Bengio, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': Large margin deep net- works for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In: NeurIPS, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 850–860 (2018) 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Ghasemi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Zahediasl, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': Normality tests for statistical analysis: a guide for non- statisticians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' International journal of endocrinology and metabolism 10(2), 486–489 (2012) 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Gal, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Ghahramani, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': Dropout as a bayesian approximation: Representing model uncer- tainty in deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' In: ICML, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 1050–1059 (2016) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Abdar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Pourpanah, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Hussain, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Rezazadegan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Ghavamzadeh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Fieguth, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Cao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Khosravi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Acharya, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Makarenkov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Nahavandi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': A review of uncer- tainty quantification in deep learning: Techniques, applications and challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Information Fusion 76(1), 243–297 (2021) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' Kendall, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=', Gal, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=': What Uncertainties Do We Need in Bayesian Deep Learning for Com- puter Vision?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content='. In: NeurIPS, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} +page_content=' 5575–5585 (2017)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfBQty/content/2301.04850v1.pdf'} diff --git a/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf b/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..4305c4f15c8658a246449ef4d05967e406e8b018 --- /dev/null +++ b/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:de8768266a5eb81d2e952d6d5a9a7411be0b9f12de2b1a232f6210c9c56b2fcc +size 678821 diff --git a/69E0T4oBgHgl3EQfwAF2/vector_store/index.faiss b/69E0T4oBgHgl3EQfwAF2/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..016455e8b939c0767dc3a81e066ff7d5b9ad132d --- /dev/null +++ b/69E0T4oBgHgl3EQfwAF2/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:bbf7c9b3745e7e7274c8990eba446184be2199a7d49fba61beca0988101661e1 +size 3145773 diff --git a/69E0T4oBgHgl3EQfwAF2/vector_store/index.pkl b/69E0T4oBgHgl3EQfwAF2/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..f6f3647654af4fcbf84820a7964fac5170898a2b --- /dev/null +++ b/69E0T4oBgHgl3EQfwAF2/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:260d6eeeff586cb584b10686a4408c8705bc2fdefa45a93cf0a19f64adae1138 +size 117366 diff --git a/7dAzT4oBgHgl3EQfgPzo/vector_store/index.faiss b/7dAzT4oBgHgl3EQfgPzo/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..7cbe3252f551bee01bc9524a7b8b4f15479b9f87 --- /dev/null +++ b/7dAzT4oBgHgl3EQfgPzo/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:302d0f4bb158aef79aa6aab404bd267136b62fc878ebf9b1e21261168497ee90 +size 5242925 diff --git a/7tE0T4oBgHgl3EQfwQFU/content/tmp_files/2301.02629v1.pdf.txt b/7tE0T4oBgHgl3EQfwQFU/content/tmp_files/2301.02629v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..54a18d246b722453a2c072f4fd81f66203658b16 --- /dev/null +++ b/7tE0T4oBgHgl3EQfwQFU/content/tmp_files/2301.02629v1.pdf.txt @@ -0,0 +1,1845 @@ +arXiv:2301.02629v1 [math.AG] 31 Oct 2022 +Intersection theory on non-archimedean analytic spaces +Yulin Cai +January 9, 2023 +Abstract +We develop the intersection theory of non-archimedean analytic spaces and prove the pro- +jection formula and the GAGA principle. As an application, we naturally define the category +of finite correspondences of analytic spaces. +Contents +1 +Introduction +1 +2 +Preliminary +3 +3 +Meromorphic functions and Cartier divisors +7 +4 +Cycles, flat pull-backs and proper push-forwards +11 +5 +Proper intersection and intersection multiplicities +19 +6 +Projection formula +22 +7 +GAGA +23 +8 +The category of finite correspondences +25 +Acknowledgements +27 +References +27 +1 +Introduction +The intersection theory of non-archimedean analytic spaces has been studied in [11, Section 2] and +[1, Section 2.2], and the author believes that some experts have concrete idea about such a theory. +In [11], Gubler considers the Cartier divisors on rigid analytic spaces and formal schemes, and +define their intersection with irreducible analytic subsets. This theory allows him to define the +local height of subvarieties over non-archimedean fields. +In [1], Ayoub develops the theory of motives on rigid analytic spaces using homotopy theory. +He uses the presheaves on the category of affinoid spaces to construct the category of finite corre- +spondence (for rigid analytic space) RigCor(K). Such construction avoids the intersection theory +of analytic spaces. +In this paper, we will develop the intersection theory of non-archimedean analytic spaces follow- +ing the idea similar to the case of algebraic varieties. We will show the flat base change formula, the +projection formula and the GAGA principle to relate the intersection theories of analytic spaces +and of algebraic varieties. As an application, we will give a direct construction of RigCor(K) +(simply denoted by CorK in this paper) like [13, Lecture 1] does. In fact, we can define the higher +Chow groups of analytic spaces as [4] for algebraic varieties, and this definition is different from +Ayoub’s in [1, Introduction g´en´erale]. +In Section 2, we give some basic notion in the theory of Berkovich spaces, e.g. support of +a coherent sheaf, Zariski image and codimension. We also extend [7, Proposition 4.12] into an +abstract form, i.e. Lemma 2.15 which is a key lemma for this paper. With this lemma, we can +solve the compatibility problems in our theory, e.g. see Lemma 4.6 and Lemma 5.4. +1 + +In Section 3, we define and study the Cartier divisors on an analytic space X, which form +a group Div(X). The group of divisors up to linear equivalence is denoted by CaCl(X). As in +the theory of schemes, we have an injective homomorphism CaCl(X) ֒→ Pic(X), and it is an +isomorphism if X is reduced. +In Section 4, we give the notion of cycles, and associate a coherent sheaf with a cycle. In +particular, we can associate a closed subspace with a cycle. As in the theory of algebraic varieties, +the flat pull-backs and proper push-forwards of cycles are defined. We prove the following flat base +change formula. +Proposition 1.1 (Proposition 4.28). Let +Y ′ +g′ +� +f ′ +� +Y +f +� +X′ +g +� X +be a Cartesian diagram of separated, strictly K-analytic spaces with f proper and g flat. Then f ′ +is proper, g′ is flat and g∗ ◦ f∗ = f ′ +∗ ◦ g′∗ on Z∗(Y ). +In Section 5, we define intersection product of proper intersection. We will give two definitions, +meaning a local one using the scheme theory and a global using Tor formula. For a flat morphism +f : Y → X of K-analytic spaces of pure dimension, the pull-back f ∗ : Z∗(X) → Z∗(Y ) preserves +intersection product. +Since we have the flat pull-backs, proper push-forwards and intersection products, the expected +projection formula is proved in Section 6. +Theorem 1.2 (Projection formula). Let f : Y → X be a flat, proper morphism of regular, +separated, strictly K-analytic spaces. Let α ∈ Z∗(Y ) and β ∈ Z∗(X). Assume that α and f ∗β +intersect properly. Then f∗(α) and β intersect properly and +f∗(α) · β = f∗(α · f ∗β). +In Section 7, we compare the intersection theories of algebraic varieties and of non-archimedean +analytic spaces. We prove the GAGA principle, i.e. Proposition 7.3. +In Section 8, we define the category of finite correspondence CorK. This category is also defined +by Ayoub [1] using another definition. +Notation and terminology +Throughout this paper, we fix a complete non-archimedean field K with a non-trivial valuation. +For a K-analytic space, we mean a Berkovich space over K, see [3, Definition 1.2.3]. The structure +sheaf on a K-analytic space X with respect to the G-topology is denoted by OX. If it is necessary, +we will use the notation XG for the G-topology instead of the ordinary topology on X. The (K- +analytic) dimension of X is denoted by dimK X, or dim X when there is no confusion with the +fields. +Given a point x ∈ X, H (x) denotes its complete residue field and dimx X denotes the local +dimension of X at x. +We shall simply say ”coherent sheaf on X” for ”coherent OX-module (with respect to G- +topology)”, and denote Pic(X) for the group of invertible sheaves on X. Assume that X is good, +let F be a coherent sheaf on X and x ∈ X. We denote by Fx the stalk at x of F viewed as a sheaf +of the underlying ordinary topology of X, i.e. +Fx := lim +−→ +U +F(U) = lim +−→ +V +F(V ). +where U runs through open neighborhoods of x, and V runs through affinoid neighborhoods of x. +We will write Irr(X) for the set of all irreducible components of X, and write Irr(X) for the +set of all irreducible Zariski-closed subsets of X. Notice that Irr(X) has a partial order: W ≤ Z if +W ⊂ Z. +2 + +For an algebraic variety over K, we mean a separated scheme of finite type over K. +For a commutative ring A, R(A) denotes the set of all regular elements of A and Frac(A) = +R(A)−1A, the maximal localization containing A as a subring. +2 +Preliminary +For the convenience of the reader and further uses, in the section, we provide some basic concepts +and results that are either given somewhere, or formulated easily. +2.1 +Support of a coherent sheaf +(cf. [8, Section 2.5]) +Definition 2.1. Let X be a K-analytic space, F be a coherent sheaf on X, and Ann(F) be the +(coherent) annihilator ideal of F (on the site XG). +The support of F is the closed analytic +subspace of X defined by Ann(F), denoted by Supp(F). +Remark 2.2. +(1) Recall the annihilator I of F is defined as follows: for any analytic domain +V , +Ann(F)(V ) := {a ∈ OX(V ) | a · F(V ) = 0}, +which is a coherent ideal. In particular, for any analytic domain V , we have Ann(F)|V = +Ann(F|V ). +(2) If X = M(A) is affinoid and F = � +M for some finitely generated A-module, then it is easy +to see that +Ann(F) = +� +Ann(M). +From the definition, we can easy deduce the following lemma. +Lemma 2.3. Let X be a K-analytic space, F a coherent sheaf on X, and Z = Supp(F). Then +there is a unique coherent sheaf G on Z such that F = i∗G, where i : Z ֒→ X is the canonical +immersion. +Proof. By uniqueness, we can glue coherent sheaf G from local parts, so we can assume that +X = M(A). It is not hard to see the lemma in this case. +2.2 +Zariski image of a morphism +As in the theory of schemes, we can define Zariski image of a morphism of analytic spaces, which +has a natural structure of analytic spaces. We follow the idea in [14, Subsection 29.6]. +Lemma 2.4. Let X be a K-analytic space, F a coherent sheaf on X, and G ⊂ F an OX-submodule. +Then there is a unique coherent OX-submodule G′ ⊂ G with the following property: for any coherent +OX-module H, the canonical map +HomOX(H, G′) → HomOX(H, G) +is bijective. In particular, G′ is the largest coherent sheaf contained in G. +Proof. Let {Gi}i∈I be the set of coherent sheaves contained in G. We consider the morphism of +OX-modules +ϕ : +� +i∈I +Gi → F. +We claim its image G′ ⊂ G is coherent. Let pG′ ⊂ G be the image of ϕ as presheaves. Then G′ is +the sheafification of pG′, and for any affinoid domain V = M(V ), pG′(V ) = � +i +Gi(V ) ⊂ F(V ) is a +finitely generated A-module. By Tate acyclic theorem, we have G′(V ) = pG′(V ). So G′ is coherent. +It is the largest coherent sheaf contained in G. +3 + +The map +HomOX(H, G′) → HomOX(H, G) +is obviously injective. For any homomorphism ψ : H → G ⊂ F, the image Im(ψ) ⊂ G is a coherent +sheaf, so Im(ψ) ⊂ G′, so f factor thorough G′. This implies that G′ is the one we want. +For the uniqueness, if G′′ is another coherent OX-submodule with the universal property. Then +the bijectivity of HomOX(G′, G′′) → HomOX(G′, G) implies that we have a homomorphism G′ → +G′′ ⊂ G, so G′ ⊂ G′′. Hence G′ = G′′. +Proposition 2.5. Let f : Y → X be a morphism of K-analytic spaces. Then there is a closed +analytic subspace Z of X such that +(a) the morphism f factors through Z; +(b) (Universal property) if f factors through a closed analytic subspace Z′ of X, then Z′ contains +Z as a closed analytic subspace. +The closed analytic space Z of X is called the Zariski image of f, denoted by Imzar(f). +Proof. By (b), if Z exists, then it is unique. It remains to show the existence. Let I := Ker(OY → +f∗OX). By Lemma 2.4, we take the largest coherent OX-submodule J ⊂ I and set Z = V (J ). It +remains to check (a) and (b). +(a) We have f(Y ) ⊂ Z. Indeed, for any affinoid domain V = M(A) ⊂ X and any affinoid +domain U = M(B) ⊂ f −1(V ), we have J (V ) ⊂ I(V ) ⊂ Ker(A → B), so U → V factors through +M(A/J (V )) = Z ∩V and f(U) ⊂ Z. Hence f(Y ) ⊂ Z. We denote the map Y → Z by f. We shall +construct f +# : OZ(V ∩ Z) → OX(f −1(V )) for any affinoid domain V ⊂ X. Since J (V ) ⊂ I(V ), +the homomorphism OX(V ) → OY (f −1(V )) factor through OZ(V ∩Z) = OX(V )/J (V ), we denote +OZ(V ∩Z) → OX(f −1(V )) by f +# which is compatible on intersections of affinoid domains. Hence +we have a morphism f : Y → Z and f = i ◦ f. +(b) If f factors through a closed subspace Z′ of X with Z′ = V (J ′), then J ′ ⊂ I. By the +choice of J , we have J ′ ⊂ J , so Z′ ⊂ Z. +Remark 2.6. +(1) Locally, f : M(B) → M(A) is given by ϕ : A → B, then Imzar(f) = +M(A/ Ker(ϕ)). +We may expect the Zariski image is exactly the usual image as sets. It is almost true if Y is +reduced or f is quasi-compact. +Lemma 2.7. Let f : Y → X be a morphism of K-analytic space. +If Y is reduced, then the +Imzar(f) = f(Y ) +Xzar with the reduce closed subspace structure. +Proof. As a map, f factor through f(Y ) +Xzar. Since Y is reduced, so f factors through f(Y ) +Xzar +with the reduced structure, see [7, PROPOSITION 4.2 (iii)]. It remains to show the universal +property of Y → f(Y ) +Xzar. If f factors through a closed subspace Z of X, then f(Y ) +Xzar ⊂ Z as +a subset. The containment is also a morphism of analytic spaces since f(Y ) +Xzar is endowed with +the reduced structure. +Lemma 2.8. Let f : Y → X be a morphism of K-analytic space. Assume that f is quasi-compact. +Then the following hold. +(1) I = Ker(OX → f∗OY ) is coherent. In particular, Imzar(f) = V (I). +(2) f(X) +Xzar = Imzar(f). In other word, Y → Imzar(f) is dominant. +(3) For any analytic domain V ⊂ X, the subspace Imzar(f)∩V is the Zariski image of f|f −1(V ) : +f −1(V ) → V . +4 + +Proof. (1) Suppose X = M(A) is affinoid. We take a G-covering Y = +n� +i=1 +Vi by affinoid domains, +and set Y ′ = +n� +i=1 +Vi, π : Y ′ → Y the canonical morphism which is surjective. For any analytic +domain V ⊂ Y , the map +π# : OY (V ) → OY ′(π−1(V )) = +n +� +i=1 +OY (V ∩ Vi) +is injective. We consider f ′ := f ◦ π : Y ′ → X. Then +I = Ker(OX → f ′ +∗OY ′). +Since Y ′ is affinoid, so I = (Ker(A → OY ′(Y ′))∼ which is coherent. This implies (1). +(3) This is from (1). +(2) By (3), suffices to assume that X = M(A) is affinoid. We use the notations in (1). Notice +that f(Y ) +Xzar = f ′(Y ′) +Xzar, so we can assume that Y = M(B) is affinoid, and f is induced by +ϕ : A → B. We have I = � +Ker(ϕ) and Imzar(f) = M(A/ Ker(ϕ)). So the morphism X → Imzar(f) +is induced by an injective homomorphism A/ Ker(ϕ) → B, hence it is dominant. +2.3 +Codimension +We recall the definition of codimension in [8, 1.5.15]. +Definition 2.9. Let X be a K-analytic space, and Y a Zariski-closed subset of X. The codimen- +sion codim(Y, X) of Y in X is defined as follows. +• If both Y and X are irreducible, codim(Y, X) := dimK X − dimK Y . +• If Y is irreducible, codim(Y, X) := +sup +Z∈Irr(X) +Y ⊂Z +codim(Y, Z). +• In the general case, codim(Y, X) := +inf +Z∈Irr(Y ) codim(Z, X). +For x ∈ X, we define the codimension of Y in X at x as +codimx(Y, X) := + + + + + +inf +Z∈Irr(Y ) +x∈Z +codim(Z, X) +if x ∈ Y ; ++∞ +if x ̸∈ Y . +Remark 2.10. +(1) Let W ⊂ Z ⊂ Y ⊂ X be irreducible closed analytic subspaces. Then +codim(W, Y ) = codim(W, Z) + codim(Z, Y ), +dimK(Z) + codim(Z, Y ) = dimK(Y ). +Example 2.11 ([6] Proposition 1.11). Let X = M(A) be a K-affinoid space, Y = V (I) for some +ideal I ⊂ A, and x ∈ X with image ξ ∈ Spec(A). Then +(1) codim(Y, X) = codim(Spec(A/I), Spec(A)). +(2) codimx(Y, X) = codimξ(Spec(A/I), Spec(A)). +Remark 2.12. +(1) In particular, (1) implies that +codim(Spec(AL/IL), Spec(AL)) = codim(Spec(A/I), Spec(A)) +for any complete field extension L/K. Or we can write +dimK X − dimK Y = codimKrull(Y, X). +5 + +Proposition 2.13. Let X be a K-analytic space, and Z, Y ∈ Irr(X) with Z ⊂ Y . Then +codim(Z, Y ) = max{m | Z = Y0 ⊊ Y1 ⊊ · · · ⊊ Ym = Y }, +where Yi ∈ Irr(X). Moreover, each maximal chain has the same length, i.e. every K-analytic space +is catenary with respect to the Zariski topology. +Proof. Firstly, if Z ⊊ Y , then codim(Z, Y ) ≥ 1. This can be seen locally. Hence ”≥” holds. +Conversely, it suffices to show that if codim(Z, Y ) ≥ 2, then there is W ∈ Irr(X) such that Z ⊊ +W ⊊ Y . Indeed, we take an affinoid domain V of Y are affinoid, and V = M(A), Z∩V = M(A/I). +Then we know that +codim(Z, Y ) = codim(Spec(A/I), Spec(A)) ≥ 2. +So we can find a prime ideal p ∈ Spec(A) such that W := M(A/p) +Yzar strictly contains Z. Apply +the same method, we can see that each maximal chain has the same length (this in fact due to the +additivity of codimension). +Remark 2.14. +(1) In particular, we see that the codimension is independent of the base field +K. +2.4 +A key lemma +For a set S satisfying certain conditions, we can determine if S satisfies a property P or not. In +this case, we say that the property P is well-defined on S. It is not well-defined if S does not +satisfy these conditions at the beginning. +The following generalized result from [7, Proposition 4.12] is crucial for extending a local result +on irreducible closed subsets to be global. +Lemma 2.15. Let X be a K-analytic space. +Let P be a property on irreducible components +satisfying the following properties: +• there is a G-covering X = � +i∈I +Vi by affinoid domain, the property P is well-defined (this +means that we can determine if P is satisfied or not) on each irreducible component of Vi (or +simply say that P is well-defined on Vi); +• if P is well-defined on an irreducible component Z of an affinoid domain V , then P is well- +defined on each irreducible component of W for any affinoid domain W ⊂ V . Moreover, in +this case, for any irreducible component T of W ∩ Z, we have T satisfies P ⇐⇒ Z satisfies +P. +Then there exist Zariski-closed subsets X+ +P , X− +P of X which are characterized by the following +properties: for any affinoid domain V on which P is well-defined, we have +X+ +P ∩ V = +� +T ∈Irr(V ), +T satisfies P +T, +X− +P ∩ V = +� +T ∈Irr(V ), +T doesn’t satisfy P +T. +Notice that X = X+ +P ∪ X− +P . +Proof. For any affinoid domain V on which P is well-defined, set +C+(V ) := {T ∈ Irr(V ) | T satisfies P}, +C−(V ) := {T ∈ Irr(V ) | T doesn’t satisfy P}, +E+(V ) := +� +T ∈C+(V ) +T, +E−(V ) := +� +T ∈C−(V ) +T. +6 + +Let V be an affinoid domain on which P is well-defined, and W ⊂ V an affinoid domain. Let Z be an +irreducible component of V and T an irreducible component of W containing Z. By our assumption, +T ∈ C+(W)⇐⇒ Z ∈ C+(V ). By [7, COROLLAIRE 4.11], we have E+(W) = E+(V ) ∩ W and +E−(W) = E−(V ) ∩ W. +Let X+ +P (resp. X− +P ) be the union of E+(V ) (resp. E−(V )) where V is an affinoid domain on +which P is well-defined. Then for any affinoid domain V of X on which P is well-defined, we +have X+ +P ∩ V = E+(V ) and X− +P ∩ V = E−(V ). Since P is well-defined on Vi for some G-covering +X = � +i∈I +Vi by affinoid domain, and E+(Vi), E−(Vi) ⊂ Vi are Zariski-closed, so X+ +P , X− +P ⊂ X are +Zariski-closed. +3 +Meromorphic functions and Cartier divisors +The sheaf of meromorphic functions and Cartier divisors are defined on a ringed space in [10, +Section 20, Section 21]. On a G-ringed space, these definitions do not work since the restriction of +a regular element is not necessarily regular. Fortunately, this can be remedied on analytic spaces +(cf. [11, Section 2]). In this section and next section, we will following the idea in [10, Section 20, +Section 21] to discuss meromorphic functions, Cartier divisors and cycles. +3.1 +Meromorphic functions +For a (commutative) ring A, denote R(A) ⊂ A the set of all regular elements, i.e. non-zero divisors, +we know R(A) is a multiplicative set, and the corresponding localization Frac(A) := R(A)−1A is +the maximal localization containing A as a subring. +Definition 3.1. Let X be a K-analytic space. For any affionid domain V = M(A) ⊂ X, we +set K′ +X(V ) := Frac(A), this will defined a presheaf on affinoid domains on X. The associated +sheaf KX with respect to the G-topology on X is called the sheaf of meromorphic functions on +X. An element of KX(X) is called a meromorphic function on X. The subsheaf of invertible +elements of KX is denoted by K∗ +X. +Remark 3.2. +(1) For affinoid domains U = M(B) ⊂ V = M(A) of X, and f ∈ R(A), the +restriction of f on U is in R(B), this implies that our definition of KX is well-defined. +Proof. It is from the fact A → B is flat, or we assume that B = A{p−1 +1 +T1,··· ,p−1 +n Tn} +(gT1−f1,··· ,gTn−fn). +(2) For any analytic domain V ⊂ X, we have +KX(V ) = + + + + + + + +(si)i ∈ +� +i +K′ +X(Vi) +�������� +V = � +i +Vi is a G-covering of V with Vi affi- +noid and si|Vijk = sj|Vijk for some G-covering +Vi ∩ Vj = � +k +Vijk with Vijk affinoid + + + + + + + +� +∼, +where (si)i ∼ (s′ +j)j if for any i, j, there exists a G-covering Vi ∩V ′ +j = � +k +Vijk with Vijk affinoid +such that si|Vijk = s′ +j|Vijk. +If X is separated, then it can be simplified as +KX(V ) = +� +(si)i ∈ +� +i +K′ +X(Vi) +����� +V = � +i +Vi is an G-covering of V with Vi affi- +noid and si|Vi∩Vj = sj|Vi∩Vj +� � +∼, +where (si)i ∼ (s′ +j)j if for any i, j, si|Vi∩V ′ +j = s′ +j|Vi∩V ′ +j . +(3) For any affinoid domain V ⊂ X, the canonical map K′ +X(V ) → KX(V ) is injective. +In +particular, OX ⊂ KX. +7 + +Proof. Given an affinoid domain V and any finite G-covering V = +n� +i=1 +Vi by affinoid domains, +let A = OX(V ) and Ai = OX(Vi). We consider the restriction map Frac(A) → +n� +i=1 +Frac(Ai). +Let a/b ∈ Frac(A) be such that its restriction on Frac(Ai) is 0 for any i, i.e. a = 0 ∈ Ai. +This implies that a = 0 ∈ A by Tate’s acyclic theorem. Hence K′ +X(V ) ֒→ KX(V ). +We take a G-covering X = � +i∈I +Vi by affinoid domains. Then the injective map OX(Vi) ֒→ +K′ +X(Vi) will induce OX ֒→ KX. +Definition 3.3. Keep the notion in Definition 3.1. For an OX-module F, we call F ⊗OX KX the +sheaf of meromorphic sections of F, and we have a canonical map +idF ⊗i : F → F ⊗OX KX. +The sheaf F is called strictly without torsion if idF ⊗i is injective. +A global section of F ⊗OX KX is called a meromorphic sections of F on X. +If F is coherent on X, we say a meromorphic section s on X is defined on a Zariski-open +subset V if s|V is in the image of F(V ) via idF ⊗i. If moreover, F is strictly without torsion, then +there is a maximal Zariski-open subset V on which s is defined, such V is called the domain of +definition of s, denoted by dom(s) (i.e. s ∈ F(dom(s))). +Remark 3.4. +(1) Notice that F → F ⊗OX KX is the sheafification of the presheaf given by +V �→ F(V ) ⊗OX(V ) K′ +X(V ) +for any affinoid domain V . So for any analytic domain V ⊂ X, we have (F ⊗OX KX)|V ≃ +F|V ⊗OV KV . In particular, KX|V = KV . +(2) A locally free OXG-module F is strictly without torsion. Moreover, F ⊗OX KX is a KX- +module, here, we view (XG, KX) as a G-ringed space. +For a good, strictly K-analytic space, the sheaf of meromorphic functions can be given in a +similar way in [10, Section 20], and will have some good properties, i.e. properties for schemes can +be extended to good analytic spaces. +If X is good, strictly K-analytic, and x ∈ X is rigid, we have that +OX,x = lim +−→ +V +OX(V ) +where V runs through affinoid domains containing x, see [2, Section 2.3]. In particular, it suffices +that V runs through (strictly) affinoid neighborhoods of x in X. +Proposition 3.5. Let X be a good, strictly K-analytic space. For any analytic domain V ⊂ X, +set +R(V ) := {s ∈ OX(V ) | sx ∈ R(OX,x) for any x ∈ V } ⊂ OX(V ), +which defines a sheaf on X. Then the following statements hold: +(1) For any affinoid domain V ⊂ X, we have R(V ) = R(OX(V )). In particular, and KX to be +the sheafification of the following presheaf: for any analytic domain V ⊂ X, +V �→ R(V )−1OX(V ). +(2) For any rigid point x ∈ X, we have K′ +X,x ≃ Frac(OX,x). For any analytic domain V ⊂ X, +the canonical homomorphism K′ +X(V ) ֒→ +� +x∈V rigid +K′ +X,x is injective. +8 + +Proof. Notice that the presheaf R is a sheaf. Since R is a subpresheaf of OX, and if V = � +i∈I +Vi is +a G-covering of an analytic domain V , ai ∈ R(Vi) such that ai|Vi∩Vj = aj|Vi∩Vj then there exists +a ∈ OX(V ) such that a|Vi = ai, then a ∈ R(V ). +(1) For any affinoid domain V ⊂ X and a ∈ OX(V ), we have a is regular ⇐⇒ a ∈ OX,x regular +for any x ∈ V . Indeed, ”=⇒” is from the flatness, for ”⇐=”, if a ∈ OX,x is regular, then there is +an affinoid neighborhood Vx of x in V such that a ∈ R(OX(Vx)) (since Ker(OX(V ) +·a +→ OX(V )) is +finitely generated). Then a ∈ R(OX(V )) since V = � +x∈V +Vx is a G-covering. So R(V ) = R(OX(V )). +Hence K′ +X(V ) = Frac(OX(V )). +(2) By definition, we have a map +lim +−→ +V +K′ +X(V ) → R−1 +x OX,x +which is surjective, where V runs through affinoid neighborhoods of x. If a/b ∈ K′ +X(V ) with V +affinoid neighborhood of x such that a/b = 0 ∈ R−1 +x OX,x, i.e. there is c ∈ Rx such that ac = 0. +We can assume that c ∈ OX(V ), then a/b = 0 ∈ K′ +X(V ). +It remains to show that Rx = R(OX,x). We have an injective map Rx ֒→ R(OX,x) by definition. +Conversely, for a ∈ R(OX,x), we consider an affinoid neighborhood V of x with A = OX(V ) such +that a ∈ A, then +0 +� Ann(a) +� A +� A . +Since Ann(a) is finitely generated and a ∈ R(OX,x), so we can find an affinoid neighborhood +U ⊂ V of x with B = OX(U) such that Ann(a) ⊗A B = 0. So a ∈ R(B). By (1), we know that +Rx = R(OX,x). +If a/b ∈ K′ +X(V ) such that 0 = a/b ∈ K′ +X,x for any rigid x ∈ V , then there exists an affinoid +neighborhood Vx of x such that 0 = a/b ∈ K′ +X(Vx). Since R(Vx) = R(OX(Vx)), we have 0 = a ∈ +OX(Vx) and a = 0 ∈ K′ +X(V ), a/b = 0. +3.2 +Cartier divisors +Definition 3.6. Let K be a complete non-archimedean field, and X a K-analytic space. We denote +the group H0(XG, K∗ +X/O∗ +X) by Div(X). The elements of Div(X) are called Cartier divisors of +XG. +Let f ∈ H0(XG, K∗ +X), its image in Div(X) is called a principal Cartier divisor and denoted +by div(f). +We say that two Cartier divisor D1, D2 are linearly equivalent if D1 − D2 is principal, write +D1 ∼ D2. We denote CaCl(X) the group of equivalent class of Cartier divisors. +A Cartier divisor D is called effective if it is in the image of the canonical map H0(XG, (OX ∩ +K∗ +X)/O∗ +X) → H0(XG, K∗ +X/O∗ +X), write D ≥ 0. The set of effective Cartier divisors is denoted by +Div+(X). +Remark 3.7. +(1) The exact sequence of sheaves +0 +� O∗ +X +� K∗ +X +� K∗ +X/O∗ +X +� 0 +will induce a long exact sequence +0 +� H0(XG, O∗ +X) +� H0(XG, K∗ +X) +� Div(X) +� +H1(XG, O∗ +X) +� H1(XG, K∗ +X) +� · · · +(2) We can represent a Cartier divisor D by a system {(Ui, fi)}i∈I, where X = � +i∈I +Ui is a G- +covering by affinoid domains, and fi = ai/bi ∈ K′ +X(Ui) such that fi|Ui∩Uj ∈ fj|Ui∩UjOX(Ui∩ +Uj)∗ for every i, j ∈ I. +Two systems {(Ui, fi)}i∈I and {(Vj, gj)}j∈J represent the same +Cartier divisor if only only if fi|Ui∩Vj ∈ gj|Ui∩VjOX(Ui ∩ Vj)∗ for any i ∈ I, j ∈ J. +9 + +If D1 = {(Ui, fi)}i∈I and D2 = {(Vj, gj)}j∈J, then D1 + D2 = {(Wijk, figj)}i∈I,j∈J, where +Ui ∩ Vj = � +k +Wijk is a G-covering by affinoid domains. +In particular, if X = M(A) is affinoid, let X = Spec(A), then we have an injection +Div(X) ֒→ Div(X). +Proposition 3.8. Keep the notion in Definition 3.6. +(1) For any divisor D = {(Ui, fi)}i∈I ∈ CaCl(X), we can associate a subsheaf OX(D) ⊂ KX +defined by OX(D)|Ui = f −1 +i +OX|Ui, which is an invertible sheaf and independent of the choice +of representative. Moreover, D ≥ 0 ⇐⇒ OX(−D) ⊂ OX. +(2) The construction above gives a homomorphism of groups ρ : Div(X) → Pic(X), +D �→ +OX(D). +(3) The homomorphism ρ induces an injective homomorphism CaCl(X) → Pic(X) with image +Im ρ = {L ∈ Pic(X) | L ֒→ KX}. +(4) If X is affinoid and reduced, then ρ : CaCl(X) → Pic(X) is an isomorphism. +Proof. We follow the idea of the proof of [12, Proposition 7.1.18]. +(1) Assume D = {(Vj, gj)}j∈J is another representative. Then +OX(D)|Ui∩Vj = f −1 +i +OX|Ui∩Vj = (gju)−1OX|Ui∩Vj = g−1 +j OX|Ui∩Vj +where u ∈ OX(Ui ∩ Vj)∗, this implies OX(D) is independent of the choice of representative. By +construction, OX(D) ∈ Pic(X), and D ≥ 0 if and only if OX(D) ⊂ OX. +(2) The map is a homomorphism. Indeed, let D1 = {(fi, Ui)}i∈I and D2 = {(gi, Ui)}i∈I, then +ρ(D1 + D2)|Ui = f −1 +i +g−1 +i +OX|Ui ≃ f −1 +i +OX|Ui ⊗OX|Ui g−1 +i +OX|Ui, +and this isomorphism is compatible on the intersection Ui ∩ Uj. +(3) If D = {(Ui, fi)}i∈I = div(f) is a principal divisor with f ∈ H0(XG, K∗ +X) and fi = f|Ui ∈ +K′ +X(Ui), where X = � +i∈I +Ui is a G-covering of X by affinoid domains. Then f −1 ∈ OX(D)(X) +because of the following exact sequence +0 +� OX(D)(X) +� � +i∈I +f −1 +i +OX(Ui) +� � +i∈I +f −1 +i +OX(Ui ∩ Uj) . +So we can define the morphism OX → OX(D), +a �→ af −1. It is an isomorphism since it is an +isomorphism on each Ui. Hence we have a homomorphism CaCl(X) → Pic(X). +If D = {(Ui, fi)}i∈I ∈ Div(X) such that OX(D) ≃ OX, then there is g ∈ OX(D)(X) such +that the morphism OX +∼ +→ OX(D), +a �→ ag is an isomorphism. Since OX(D)|Ui ≃ f −1 +i +OX|Ui = +g|UiOX|Ui and f −1 +i +∈ K′∗ +X(Ui), g|Ui = f −1 +i +ui ∈ K′∗ +X(Ui) ⊂ K∗ +X(Ui) with ui ∈ O∗ +X(Ui), we have +g ∈ K∗ +X(X) and D = {(Ui, fi)}i∈I = {(Ui, g−1|Ui)}i∈I is principal. +By definition, we know that OX(D) ⊂ KX. Conversely, for L ∈ Pic(X) with L ⊂ KX, there is +G-covering X = � +i∈I +Ui by affinoid domains such that OX|Ui ≃ L|Ui. We take gi ∈ L(Ui) which is +mapped to 1. Then gi ∈ KX(Ui) and L|Ui = giOX|Ui, moreover, there is fi ∈ K∗ +X(Ui) such that +figi = 1 because of the isomorphism. On Ui ∩ Uj, we have +L|Ui∩Uj = f −1 +i +OX|Ui∩Uj = f −1 +j +OX|Ui∩Uj, +so there is u ∈ O∗ +X(Ui ∩ Uj) such that f −1 +i +|Ui∩Uj = uf −1 +j +|Ui∩Uj. +Then L = OX(D), where +D = {(Ui, fi)}i∈I ∈ Div(X). +10 + +(4) Let X = Spec(OX(X)), then CaCl(X) ≃ Pic(X), see [12, Corollary 1.19]. We a commuta- +tive diagram +Div(X)� � +� +∼ +� +Div(X) +ρ +� +Pic(X) +∼ +� Pic(X) +, +so our claim holds. The isomorphism Pic(X) ≃ Pic(X) is from Coh(X) ≃ Coh(X) and Tate’s +acyclic theorem, see the proof of [3, Propostion 1.3.4 (iii)]. +Remark 3.9. +(1) We know that H1(XG, O∗ +X) ≃ Pic(X), then ρ is the connecting map of the +long exact sequence. +Example 3.10. Let L be a line bundle on a normal K-analytic space X. Let s ∈ H0(X, L⊗OX KX) +be a rational section which is non-zero on each irreducible component. Let X = � +i∈I +Ui be a G- +covering of X by integral affinoid domains such that L|Ui is free and generated by an element ei. +Then these exist fi ∈ K∗ +X(Ui) such that s|Vi = fiei. Moreover div(s) := {(Ui, fi)}i∈I is a Cartier +divisor such that OX(div(s)) ≃ L. +3.3 +Inverse image of a Cartier divisor +Next we consider the restriction of Cartier divisors on a closed analytic subspace. +Definition 3.11. Let D ∈ Div(X), and Z ∈ Irr(X) with reduced analytic space structure.We +say D intersects Z properly if there is a G-covering X = � +i∈I +Ui by affinoid domains such that +D = {(Ui, ai/bi)}i∈I with the images ai, bi ∈ R(OZ(Ui∩Z)). The set of Cartier divisor intersecting +Z properly is a subgroup of Div(X), denoted by GZ/X. +Remark 3.12. +(1) There is a natural homomorphism GZ/X → Div(Z) denoted by D �→ D|Z, +compatible with the homomorphism OX → i∗OZ. Moreover, we have a canonical isomor- +phism OX(D)|Z ≃ OZ(D|Z). +4 +Cycles, flat pull-backs and proper push-forwards +4.1 +Cycles +Definition 4.1. Let X be a K-analytic space. A prime cycle on X is an element in Irr(X). A +cycle on X is a formal sum α = +� +Z∈Irr(X) +nZ[Z] with nZ ∈ Z which is G-locally finite, i.e. the set +{Z ∈ Irr(X) | Z ∩ V ̸= ∅, nZ ̸= 0} +is finite for any affinoid domain V . The coefficient nZ is called the multiplicity of α at Z, +denoted by multZ(α). We say that a cycle α is positive if multZ(α) ≥ 0 for any Z ∈ Irr(X). The +set of cycles (resp. positive cycles) is denoted by Z(X) (resp. Z+(X)). +The union of the Z such that nZ ̸= 0 is called the support of α, denoted by Supp(α). It is a +Zariski-closed subset of X. By convention, Supp(0) = ∅. +A cycle α is (purely) of codimension r (resp. of dimension r) if any Z ∈ Irr(X) with +nZ ̸= 0 has codimension r (resp. dimension r). The cycles of codimension r (resp. of dimension +r) form a subgroup Zr(X) (resp. Zr(X)) of the group of cycles on X. +Remark 4.2. +(1) For a positive cycle α = +� +Z∈Irr(X) +nZ[Z] and any Z ∈ Irr(X) with nZ ≥ 1, we +can endow Z with the reduced subscheme structure, then Z = V (IZ) is an integral closed +analytic subspace of X, where IZ is the coherent sheaf of ideal defining Z. We view α as +a closed analytic subspace defined by the sheaf of ideal Iα := +� +Z∈Irr(X) +InZ +Z +and we have a +11 + +canonical closed immersion j : α = V (Iα) ֒→ X. This induces a homomorphisms of semi- +groups +Z+(X) → {closed analytic subspace of X} = {coherent sheaves of ideals on X}. +(2) By Proposition 2.13, we know that Zr(X) is not dependent of the base field K, but Zr(X) is. +Example 4.3. Let X = M(A) be a K-affinoid space. Set X = Spec(A). Then +Div(X) ֒→ Div(X), +Z∗(X) ≃ Z∗(X). +The first arrow is also an isomorphism if X is regular, see Proposition 4.13. +Lemma 4.4. Let X be a K-analytic space. Let α ∈ Z+(X) with associated sheaf of ideal Iα. Then +V (Iα) = Supp(α) with Irr(V (Iα)) = {maximal elements in α}. +Proof. This is local, and we can deduce this lemma from the example above. +The following lemma is obvious. +Lemma 4.5. Let X = � +i∈I +V be a G-covering of by affinoid domains, and α, β ∈ Div(X) (resp. +Z∗(X)). Then α = β ⇐⇒ α|Vi = β|Vi for any i ∈ I. +Proof. It suffices to show the ”if” part. If α, β ∈ Div(X), then the result holds from the expression +of Cartier divisors. If α = � +Z +nZ[Z], β = � +Z +mZ[Z] ∈ Zk(X) such that α|Vi = β|Vi for any i ∈ I, +then nZ[Z ∩ Vi] = mZ[Z ∩ Vi] for any Z ∈ Irr(X) with Z ∩ Vi ̸= ∅, so nZ = mZ. +4.2 +Cycle associated to a coherent sheaf +We will construct a homomorphism Div(X) → Z1(X) as we do in algebraic geometry. Recall, +for a Noetherian affine scheme X = Spec(A), a coherent sheaf F = � +M on X, and an irreducible +component Z of Supp(F), we set multZ(F) := lengthAp(Mp), called the multiplicity of Z in F, +where p ∈ X is the prime ideal corresponding to Z. For a divisor D ∈ Div(X) and a codimension +one prime cycle Z = {z} ∈ Z1(X), we set multZ(D) := multOX,z(Dz) the multiplicity of Z in D. +For an affinoid space M(A), we have similar notation. +Lemma 4.6. Let X be a K-analytic space. Let F be a coherent sheaf on X. For any irreducible +component Z of Supp(F) with reduced analytic space structure, and an affinoid domain V ⊂ X +with Z ∩ V ̸= ∅, we set +multZ(F) := multT (F|V ) +where T is an irreducible component of Z ∩ V with T +Supp(F)Zar = Z. Then multZ(F) is a positive +integer which is independent of the choice of T and V . We call multZ(F) the multiplicity of Z +in F. +Proof. For a fixed irreducible component Z of Supp(F), and any affinoid domain V, W ⊂ X with +W ⊂ V , Z ∩ W ̸= ∅, we claim that +multT (F|V ) = multT ′(F|W ) +where T ∈ Irr(Z ∩ V ) (resp.T ′ ∈ Irr(Z ∩ W)) with T ′VZar = T , T +XZar = Z. Indeed, let V = +M(A), W = M(B) and F|V = � +M. We shall show that +lengthAp(Mp) = lengthBq(Mp ⊗Ap Bq) +where p ⊂ A (resp. q ⊂ B) is the prime ideal corresponding to T (resp. T ′). Notice that the kernel +W → Spec(B) is surjective, we can find a y ∈ W such that Ker(| · |x) = q. Let x ∈ V be the image +of y, then Ker(| · |x) = p. We have H (x) = H (y) and +lengthAp(Mp) = dimk(p)(M ⊗A k(p)) = dimH (x)(M ⊗A H (x)), +12 + +it is similar for lengthBq(Mp ⊗Ap Bq). Hence our claim holds. +To show the lemma, we apply Lemma 2.15. +Let Z ∈ Z1(X) be a prime cycle, and m = +multT (F|V ) for some affinoid domain V ⊂ X with Z ∩ V ̸= ∅, where T ∈ Irr(Z ∩ V ) with +T +XZar = Z. For V given as before, we say an irreducible component T ∈ Irr(Z ∩ V ) satisfies P if +multT (F|V ) = m. After replacing X by Z, from our claim, we see that P satisfies the hypothesis +in Lemma 2.15. Then there are Zariski-closed subsets Z+ +P , Z− +P of Z such that +Z+ +P ∩ V = +� +T ∈Irr(Z∩V ), +T satisfies P +T, +Z− +P ∩ V = +� +T ∈Irr(Z∩V ), +T doesn’t satisfy P +T, +and Z = Z+ +P ∪ Z− +P . Since Z is irreducible and there is some T ⊂ Z+ +P , we have Z = Z+ +P . This +implies the lemma. +Definition 4.7. Keep the notion in Lemma 4.6. For a coherent sheaf F on X with codim(Supp(F), X) ≥ +k, we set +[F]k := +� +Z∈Irr(Supp(F))k +multZ(F)[Z] ∈ Zk(X), +called the cycle associated to F with codimension k. +Remark 4.8. +(1) By Lemma 4.6, it is hard to have the following result. Let V = M(A) ⊂ X +is an affinoid domain, and F a coherent sheaf on X. Set V = Spec(A) and Fal +V the coherent +sheaf on V corresponding to F|V . Then +[F|V ]k = [Fal +V ]k, +here we identify Z∗(V ) ≃ Z∗(V). +Definition 4.9. Keep the notion in Lemma 4.6. +For a closed analytic subspace Y of X with +codim(Y, X) ≥ k, we set +multZ(Y ) := multZ(OY ), +for any Z ∈ Irr(Y ), called the multiplicity of Z in Y , and set +[Y ]k := +� +Z∈Irr(Y ) +Z∈Zk(X) +multZ(Y )[Z] ∈ Zk(X), +called the cycle associated to Y with codimension k. +4.3 +Weil divisors +Definition 4.10. Let X be a K-analytic spaces. An element in Z1(X) is called a Weil divisor +on X. +Lemma 4.11. Let X be a K-analytic space. Let D ∈ Div(X). For any prime cycle Z ∈ Z1(X), +and any affinoid domain V ⊂ X with Z ∩ V ̸= ∅, D|V ∈ K′ +X(V ), we set +multZ(D) := multT (D|V ) +where T ∈ Irr(Z ∩ V ) with T +XZar = Z. Then multZ(D) is independent of the choice of T and V . +We call multZ(D) the multiplicity of Z for D. +13 + +Proof. The proof is similar with the one of Lemma 4.6. +For any prime cycle Z ∈ Z1(X) and any affinoid domain V, W ⊂ X with W ⊂ V , Z ∩ W ̸= ∅, +D|V ∈ K′ +X(V ), we claim that +multT (D|V ) = multT ′(D|W ), +where T ∈ Irr(Z ∩V ) (resp.T ′ ∈ Irr(Z ∩W)) with T ′VZar = T , T +XZar = Z. Indeed, since both sides +are additive, we can assume that D|V = f ∈ R(OX(V )). Let Y ⊂ V be a closed analytic subspace +determined by f ∈ OX(V ), then our claim is from Lemma 4.6. +To show the lemma, we apply Lemma 2.15. Let m = multT (D|V ) for some affinoid domain +V ⊂ X with Z ∩ V ̸= ∅, D|V ∈ K′ +X(V ), where T ∈ Irr(Z ∩ V ) with T +XZar = Z. For V given +as before, we say an irreducible component T ∈ Irr(Z ∩ V ) satisfies P if multT (D|V ) = m. After +replacing X by Z, from our claim, we see that P satisfies the hypothesis in Lemma 2.15. Then +there are Zariski-closed subset Z+ +P , Z− +P of Z such that +Z+ +P ∩ V = +� +T ∈Irr(Z∩V ), +T satisfies P +T, +Z− +P ∩ V = +� +T ∈Irr(Z∩V ), +T doesn’t satisfy P +T, +and Z = Z+ +P ∪ Z− +P . Since Z is irreducible, and there is some T ⊂ Z+ +P , so Z = Z+ +P . This implies +the lemma. +Definition 4.12. Let X be a K-analytic space. For any D ∈ Div(X), we set +[D] := +� +Z⊂Irr(X) +codim(Z,X)=1 +multZ(D)[Z] ∈ Z1(X), +called the Weil divisor associated to D. In particular, for any f ∈ K∗(X), we denote (f) := +[div(f)] ∈ Z1(X). Such a divisor (f) is called a principal divisor. The set of principal divisors +Rat1(X) form a subgroup of Z1(X). We denote the quotient of Z1(X) by the subgroup of principal +divisors by Cl(X) := Z1(X)/Rat1(X), called the class group of X. We say that two divisors +Z, Z′ are rationally equivalent and write Z ∼rat Z′ if they have the same class in Cl(X). +Recall, a K-analytic space X is regular at x ∈ X if there is a good analytic domain V of X +containing x such that OV,x is regular. We say X is regular if X is regular at every point x ∈ X. +This is equivalent to that for any affinoid domain V ≃ M(A) ⊂ X, we have that A is regular, see +[8, Lemma-Definition 2.4.1, Lemma 2.4.5]. +Proposition 4.13. The map [·] : Div(X) → Z1(X) a homomorphism which sends effective divisors +to positive cycles. This induces a homomorphism +[·] : CaCl(X) → Cl(X). +If X is normal (resp. regular), then these two map are injective (resp. isomorphic). +Proof. It is easy to see that [·] : Div(X) → Z1(X) is a homomorphism and induces [·] : CaCl(X) → +Cl(X). If X is normal, by Lemma 4.5, to show [·] : Div(X) → Z1(X) is injective, we can assume X +is affinoid. For D ∈ Div(X) such that multZ(D) = 0 for any Z ∈ Z1(X), we take affinoid domain +V ⊂ X with Z ∩ V ̸= ∅ and D|V ∈ K′ +X(V ). Then D|V ∈ O∗ +X(V ) since multT (D|V ) = 0 for any +Q ∈ Z1(V ). This implies that D = 0. As for the quotient, if [D] = (f) for some f ∈ K∗ +X(X), then +D = div(f), this implies that [·] : CaCl(X) → Cl(X) is injective. +Assume that X is regular. To show that [·] : Div(X) → Z1(X) is surjective, we firstly assume +that X = M(A) is affinoid and set X = Spec(A). +In this case, Div(X) ≃ Z1(X), see [12, +Proposition 7.2.16]. Hence, we have a commutative diagram +Div(X)� � +� +∼ +� +Div(X) +ρ +� +Z1(X) +∼ +� Z1(X) +, +14 + +so our claim holds for affinoid spaces. We can glue Cartier divisors on affinoid domains together +by injectivity of [·]. Hence [·] : Div(X) → Z1(X) is surjective. +4.4 +Rational equivalence of cycles +As in the classical definition of Chow group of an algebraic variety, we can extend the class group +for any codimension. +Definition 4.14. Let X be a K-analytic space. For any (k + 1)-dimensional irreducible closed +analytic subspace W of X and any f ∈ K∗ +W (W), we have a k-cycle [div(f)] ∈ Zk(W) ⊂ Zk(X) +given in Definition 4.12. A k-cycle α is rationally equivalent to zero, write α ∼ 0, if there are +a finite number of (k + 1)-dimensional subvarieties Wi of X, and fi ∈ K∗ +Wi(Wi) such that +α = +� +i +[div(fi)]. +Since [div(f −1)] = −[div(f)], the cycles rationally equivalent to zero form a subgroup Ratk(X) ⊂ +Zk(X). +The group of k-cycles modulo rational equivalence on X is the quotient +Ak(X) := Zk(X)/Ratk(X). +Define Z∗(X) (resp. A∗(X)) to be the direct sum of the Zk(X) (resp. Ak(X)) for k ∈ Z. A cycle +class on X is an element of A∗(X). +A cycle class is positive if it can be represented by a positive cycle. +Remark 4.15. +(1) The subgroup Ratk(X) ⊂ Zk(X) is well-defined by Lemma 2.13. +(2) Ak(X) = Ak(Xred) for any k ∈ Z. +(3) If X is of pure dimension n, then An(X) = Zn(X) is the free abelian group generated by the +irreducible components of X. +4.5 +Flat pull-backs +We have introduced Cartier divisors, cycles. Next we consider their pull-backs via flat morphisms. +Recall the definition of flatness in sense of [8, Definition 4.1.8], a morphism f : Y → X of K- +analytic spaces is naively flat if for any y ∈ Y , there exist a good analytic domain V ⊂ Y containing +y and a good analytic domain U ⊂ X containing f(V ) such that OV,y is flat over OU,f(y). We say +f is flat if moreover Y ′ := Y ×X X′ → X′ is naively flat for any morphism X′ → X. If f is flat, +then OY (V ) is flat over OX(U) for any affinoid domains V ⊂ Y and U ⊂ X with f(V ) ⊂ U. The +converse is not true in general unless f is locally finite. Notice that for any analytic domain V of +X, the natural morphism V ֒→ X is flat. +Definition 4.16. A morphism f : Y → X of K-analytic spaces has relative dimension r if for +any Z ∈ Irr(X), f −1(Z) = ∅ or any irreducible component Z′ of f −1(Z) has dimK Z′ = dimK Z+r. +Remark 4.17. +(1) The notion of relative dimension r is an analogue of the one in algebraic +geometry, see [9, B.2.5]. Our definition is different from the one in [8, 1.4.13]. We don’t +assume that such morphisms are surjective. +Lemma 4.18. Let f : Y → X be a flat morphism of K-analytic spaces. Then f has relative +dimension r if and only if Yx = ∅ or Yx is of equidimension r for any x ∈ X. In particular, if +f : Y → X is flat with X, Y equidimensional, then f has relative dimension dimK Y − dimK X. +Proof. We apply [8, Lemma 4.5.11] saying that dimy Y = dimy Yx + dimx X for any y ∈ Yx. +Assume that f has relative dimension r. If x ∈ X such that Yx ̸= ∅, then for any Z ∈ Irr(X) +containing x, we have dimK f −1(Z)−dimK Z = r. This implies that dimy Yx = dimy Y −dimx X = +r for any y ∈ Yx since dimx X = +max +x∈Z∈Irr(X){dimK Z}. +15 + +Conversely, for any Z ∈ Irr(X) with f −1(Z) ̸= ∅, without loss of generality, we can assume +that Z = X. We take y ∈ Y and x = f(y). Then dimy Y = dimx X + dimy Yx = dimK X + r. This +implies that f has relative dimension r. +If X, Y are equidimensional, then dimy Yx = dimy Y − dimx X implies that Yx is of equidimen- +sion for any y ∈ Y, x = f(x). +Definition 4.19. Let f : Y → X be a flat morphism of K-analytic spaces. +(1) The canonical morphism f # : OX → f∗OY extends to a morphism f # : K∗ +X/O∗ +X → +f∗(K∗ +Y /O∗ +X), then we have a homomorphism +f ∗ : Div(X) → Div(Y ). +This will induce a homomorphism f ∗ : CaCl(X) → CaCl(Y ). +(2) Assume that X, Y are of equidimension. For any integral closed subspace Z ⊂ X of pure +codimension k, we set +f ∗[Z] := [f −1(Z)] ∈ Zk(Y ). +This extends by linearity to a pull-back homomorphism f ∗ : Zk(X) → Zk(Y ). +Remark 4.20. +(1) The flat pull-backs are functorial and we have a commutative diagram +Div(X) +f ∗ +� +[·] +� +Div(Y ) +[·] +� +Z1(X) +f ∗ +� Z1(Y ) +. +Proposition 4.21. Let f : Y → X be a flat morphism of K-analytic spaces of pure dimension. +For a coherent sheaf F on X with codim(Supp(F), X) ≥ k, we have codim(Supp(f ∗F), X) ≥ k +and +[f ∗F]k = f ∗[F]k. +In particular, if Z is a closed analytic subspace of X of pure codimension k, then f ∗[Z] = [f −1(Z)]. +Proof. We can reduce the statement to the case of affinoid spaces by Lemma 4.5, then the proposi- +tion from the analogue result in scheme theory by Remark 4.8 (1). For the result in scheme theory, +see proof of [14, Lemma 42.14.4 (2)]. +4.6 +Proper push-forward of cycles +For an affinoid space X = M(A), it may happen that dimKrull A < dimK X. In order to avoid +this dimension problem, we assume that all K-analytic spaces (including affinoid domains) in this +subsection are strict. In this case dimKrull A = dimK X. +Recall a theorem of Kiehl. +Theorem 4.22 ([2] Proposition 3.3.5). Let f : Y → X be a proper morphism of K-analytic spaces, +and F a coherent OY -module. Then Rnf∗F, n ≥ 0, are coherent OX-modules. In particular, we +have Remmert’s mapping theorem, saying that f(Y ) is an Zariski-closed subset of X. +A similar result of the following lemma is given in [11, 2.6]. +Lemma 4.23. Let f : Y → X be a surjective finite morphism of integral, strictly K-analytic +spaces. For any (strictly) affinoid domain V ⊂ X and T ∈ Irr(V ), we set +deg(Y/X) := +� +Q∈Irr(f −1(V )) +f(Q)=T +[Frac(AQ) : Frac(AT )], +where AT , AQ are the affinoid algebras corresponding to T, Q with reduced structure. Then deg(Y/X) +is independent of the choice of V and T , called the degree of f. +16 + +Proof. Apply the usual technique with Lemma 2.15, it is sufficient to show that for any affinoid +domain V, W ⊂ X with W ⊂ V , and any T ∈ Irr(V ), T ′ ∈ Irr(W), we have +� +Q∈Irr(f −1(V )) +f(Q)=T +[Frac(AQ) : Frac(AT )] = +� +Q′∈Irr(f −1(W)) +f(Q′)=T ′ +[Frac(AQ′) : Frac(AT ′)]. +This is in fact from Lemma 4.6 and Proposition 4.21 for affinoid case. Let V = M(A), f −1(V ) = +M(B) and W = M(A′), then f −1(W) = M(B′), where B′ = A′⊗AB. Let F be the corresponding +coherent sheaf associated to B as an A-module on V , and i : W → V the canonical morphism, +then +[F]0 = +� +T ∈Irr(V ) +( +� +Q∈Irr(f −1(V )) +f(Q)=T +[Frac(AQ) : Frac(AT )])[T ], +and we know that +� +Q∈Irr(f −1(V )) +f(Q)=T +[Frac(AQ) : Frac(AT )] is independent of the choice of T by Lemma 4.6. +We also have +i∗[F]0 = +� +T ∈Irr(V ) +( +� +Q∈Irr(f −1(V )) +f(Q)=T +[Frac(AQ) : Frac(AT )]) +� +T ′∈Irr(T ∩W) +[T ′], +[i∗F]0 = +� +T ′∈Irr(W) +( +� +Q′∈Irr(f −1(W)) +f(Q′)=T ′ +[Frac(AQ′) : Frac(AT ′)])[T ′]. +By Proposition 4.21, we compare the coefficient of some for any irreducible component T ′, we can +see that our claim holds. +We have the following equivalent conditions. +Lemma 4.24. Let f : Y → X be a morphism of integral, separated, strictly K-analytic spaces. +Then the following are equivalent. +(i) f is surjective and finite. +(ii) f is surjective, proper, and dimK Y = dimK X. +(iii.a) f is proper, and for any x ∈ X, dimH (x) f −1(x) = 0. +(iii.b) f is proper, and for any rigid point x ∈ X, f −1(x) ̸= ∅ has finite rigid points as an H (x)- +analytic space. +(iv.a) f is surjective and proper, and there is a point x ∈ X such that dimH (x) f −1(x) = 0. +(iv.b) f is surjective and proper, and there is a rigid point x ∈ X such that dimH (x) f −1(x) = 0, +i.e. f −1(x) ̸= ∅ and has finite rigid points. +Proof. Obviously, (i) =⇒ (iii.a), (iii.b) =⇒ (iv.b) =⇒ (iv.a). +(iii.a) =⇒ (ii). This is from [8, 1.4.14 (3)]. +(ii) =⇒ (iii.b). Since f is quasi-compact, after taking irreducible components of affinoid domain +of X, Y , we can assume that X = M(A), Y = M(B) are affinoid, integral and dim A = dim B. +Moreover, since the original morphism is surjective, we know that the corresponding morphism +ϕ : Spec(B) → Spec(A) is dominant. For any closed point x ∈ Spec(A) with ϕ−1(x) ̸= ∅, by basic +property of strict affinoid algebras, we know that codim(x, Spec(A)) = dim A. Since ϕ is dominant, +then dim B ≥ codim(x, Spec(A)) + dim ϕ−1(x). So dim ϕ−1(x) = 0. Notice that K → A → H (x) +is finite, then H (x) is the residue field of Spec(A) at x, and B ⊗A H (x) = B �⊗AH (x). Hence +the rigid points of f −1(x) is exactly the closed points of ϕ−1(x) which are finite since B �⊗AH (x) +Noetherian. +17 + +(iii.b) =⇒ (i). The separatedness ensure that X, Y are also rigid K-analytic spaces, see [3, +Theorem 1.6.1]. Then the result is from [5, Corollary 9.6.6] and [2, Proposition 3.3.2]. +(iv.a) =⇒ (ii). Notice that we have proved the equivalence (i) ⇐⇒ (ii) ⇐⇒ (iii.a) ⇐⇒ (iii.b). +By [6, TH´EOR`EME 4.9], the set +{y ∈ Y | dimy f ≥ 1} +is Zariski-closed in Y . So +{x ∈ X | dimH (x) f −1(x) ≥ 1} = f({y ∈ Y | dimy f ≥ 1}) +is Zariski-closed in X, i.e. U := {x ∈ X | dimH (x) f −1(x) ≤ 0} is Zariski-open in X. Then +dimK f −1(U) = dimK U by the equivalence of (iii.a) and (ii). Since dimK Y = dimK f −1(U), dimK X = +dimK U, we have (ii). +With the lemmas above, we have the following definition. +Definition 4.25. Let f : Y → X be a proper morphism of separated, strictly K-analytic spaces. +For any irreducible closed subspace Z of Y , the image f(Z) is a Zariski-closed subset of Y . We set +deg(Z/f(Z)) := +� +the degree of f : Z → f(Z) +if dimK f(Z) = dimK Z; +0 +if dimK f(Z) < dimK Z +(notice that dimK f(Z) = dimK Z is equivalent to f : Z → f(Z) is finite). +Define f∗[Z] := +deg(Z/f(Z))[f(Z)], then extends linearly to a homomorphism (of gradding groups) +f∗ : Z∗(Y ) → Z∗(X). +Remark 4.26. +(1) For Z above, we know that f(Z) with the reduced subspace structure is the +Zariski image of Z → X by Lemma 2.7. +We can easily prove the following lemma. +Lemma 4.27. Let f : Y → X and g : Z → Y be proper morphism of separated strictly K-analytic +spaces. Then g∗ ◦ f∗ = (g ◦ f)∗. +Proposition 4.28. Let +Y ′ +g′ +� +f ′ +� +Y +f +� +X′ +g +� X +be a Cartesian diagram of separated, strictly K-analytic spaces with f proper and g flat. Then f ′ +is proper, g′ is flat and g∗ ◦ f∗ = f ′ +∗ ◦ g′∗ on Z∗(Y ). +Proof. The morphism f ′ is proper by [3], and g′ is flat by definition. +For the equality, notice that it holds if f is a closed immersion. In general, To show g∗(f∗α) = +f ′ +∗(g′∗(α)), we can assume that α = [Y ] and it is irreducible. +Moreover, we can assume that +X = f(Y ). +If dimK X < dimK Y , then left-handed side is 0. For any x′ ∈ X′, let x = g(x′). We have +(f ′)−1(x′) = M(H (x′)) ×X′ Y ′ = M(H (x′)) ×X Y = M(H (x′)) ×H (x) f −1(x). +Since f is not finite, by Lemma 4.24 (iv.a), we have dimH (x′)(f ′)−1(x) = dimH (x) f −1(x) > 0. +This means that f ′ is not finite, and f ′∗([Y ′]) = 0. +If dimK X = dimK Y , then f : Y → X is finite. By Lemma 4.5, it suffices to consider the affine +case. Then the result is from Proposition 4.21, and can be proved similarly as Lemma 4.23. +With the proposition above, we can always assume that the base space is affinoid. We can +use this to deduce the following result to the scheme case, see [14, Lemma 42.12.4] for the scheme +version. +18 + +Proposition 4.29. Let f : Y → X be a proper morphism of separated strictly K-analytic spaces. +(1) Let Z ⊂ Y be a closed subspace with dimK Z ≤ k. Then +f∗[Z]k = [f∗OZ]k. +(2) Let F be a coherent sheaf on X such that dimK(Supp(F)) ≤ k. Then +f∗[F]k = [f∗F]k. +Proof. Obviously, it suffices to show (2). By Lemma 2.3, there is a coherent sheaf G on Z := +Supp(F) such that F = i∗G. Let Z′ be the Zariski image of Z → X. Notice that f(Z) = Z′ by +Lemma 2.8 and properness of f. So we have the following commutative diagram +Z� � +� +f|Z � +Y +f +� +Z′� � +� X +. +By functorial property of push-forward, it suffices to show (f|Z)∗[G] = [(f|Z)∗G]. So we can assume +that dimK X = k and f : X → Y is proper and dominant. Moreover, we can assume that Y is +affinoid. So dimK Y ≤ k. +We write +f∗[F]k = +� +W +nW [W] +and +[f∗F]k = +� +W +mW [W] +where W runs through irreducible component of X of dimension k. For a fixed irreducible com- +ponent W, to show nW = mW , it suffices to show that (f∗[F]k)|V = ([f∗F]k)|V for some affinoid +domain V ⊂ X with V ∩ W ̸= ∅. We can take Zariski-open subsets U ⊂ X such that U ∩ W ′ = ∅ +and U ∩ f(T ) = for any irreducible component W ′ of X which is distinct from W, and any ir- +reducible component T of Y which doesn’t dominate W. We can take an affinoid domain of U. +So we can assume X = M(A) is equidimensional and each irreducible component of Y dominates +some irreducible component of X. By [2, Corollary 3.3.8], we know that Y is finite over X. So we +reduce to the case where Y, X is affinoid and f is finite. This is an algebraic result, see the last +part of the proof of [14, Lemma 41.13.3]. +5 +Proper intersection and intersection multiplicities +5.1 +Proper intersection +Lemma 5.1. Let X be a regular K-analytic space of pure dimension, and Y, �Y ∈ Irr(X). Then +for every irreducible component Z of Y ∩ �Y , we have +codim(Z, X) ≤ codim(Y, X) + codim(�Y , X). +Proof. The proof is based on the corresponding result in scheme theory. We can assume that X +is irreducible. For any affinoid domain V ⊂ X, we have codim(T, V ) = codim(Y, X), where T is a +irreducible component of V ∩Y . Then we can apply the corresponding result in scheme theory. +Definition 5.2. Let X be a regular K-analytic space of pure dimension. +(1) Let Y, �Y ∈ Irr(X). We say that Y and �Y intersect properly if codim(Z, X) ≥ codim(Y, X)+ +codim(�Y , X). +(2) Let α = � +i∈I +ni[Yi] ∈ Zs(X) and β = � +j∈J +mj[�Yj] ∈ Zr(X). We say that α and β intersect +properly if Yi and �Yj intersect properly for all i and j. +19 + +Lemma 5.3. Let X be a regular K-analytic space of pure dimension, and Y, �Y ∈ Irr(X). Then +the following statements are equivalent: +(i) Y, �Y intersect properly; +(ii) For any x ∈ Y ∩ �Y , there is an affinoid domain V containing x such that any Q ∈ Irr(Y ∩ +V ), �Q ∈ Irr(�Y ∩ V ) intersect properly on V ; +(iii) For any affinoid domain V with Y ∩ V , �Y ∩ V ̸= ∅ and any Q ∈ Irr(Y ∩ V), �Q ∈ Irr(�Y ∩ V ), +we have Q and �Q intersect properly. +Proof. For any affinoid domain V ⊂ X with Y ∩ V = ∅ and any Q ∈ Irr(Y ∩ V ), we have +codim(Q, V ) = codim(Y, X). Then the lemma follows. +5.2 +Multiplicities and intersect products +In this subsection, we will apply the intersection theory on a regular catenary Noetherian scheme +to define multiplicities. Another definition using Tor formula will be given in the next subsection. +Recall, on a regular, catenary Noetherian scheme X, let Q, �Q be irreducible closed subschemes +with codim(Q, X) = s, codim( �Q, X) = t. Then intersection product of Q, �Q is defined by +Q · �Q = +� +T +eT[T ] := +� +i +(−1)i[TorOX +i +(OQ, O � +Q)]s+t ∈ Zs+t(X), +i.e. +eT = e(X, Q · �Q, T ) = +� +i +(−1)ilengthOX,T (TorOX,T +i +(OQ,T , O � +Q,T )) +where T runs through Irr(Q ∩ �Q) with codim(T, X) = s + t, and OX ,T (resp. OQ,T , resp. O � +Q,T ) +denotes the local ring of X (resp. Q, resp. �Q) at the generic point of T . +Lemma 5.4. Let X be a regular K-analytic space of pure dimension, and Y, �Y ⊂ X irreducible +Zariski-closed subspaces with codim(Y, X) = s, codim(�Y , X) = t. Assume that Y and �Y intersect +properly. For any irreducible component Z of Y ∩ �Y with codim(Z, X) = s + t, and any affinoid +domain V ⊂ X with Z ∩ V ̸= ∅, we set +e(X, Y · �Y , Z) := +� +Q, � +Q +e(V, Q · �Q, T ) +where T ∈ Irr(Z ∩ V ) and (Q, �Q) runs through Irr(Y ∩ V ) × Irr(�Y ∩ V ) such that T ∈ Irr(Q ∩ �Q). +Then e(X, Y, �Y , Z) is a positive integer which is independent of the choice of V and T . We call +e(X, Y, �Y , Z) the multiplicity of Z on Y ∩ �Y . +Proof. The idea of proof is similar with the proof of Lemma 4.6 and Lemma 4.11. It is sufficient +to show that for any affinoid domain V, W ⊂ X with W ⊂ V , Z ∩ W ̸= ∅, we have that +� +Q, � +Q +e(V, Q · �Q, T ) = +� +Q′, � +Q′ +e(W, Q′ · �Q′, T ′) +where T ∈ Irr(Z ∩ V ), (Q, �Q) runs through Irr(Y ∩ V ) × Irr(�Y ∩ V ) such that T ∈ Irr(Q ∩ �Q), +and T ′, Q′, �Q′ is given similarly with T ′VZar = T , T +XZar = Z. Let V = M(A), W = M(B) and +f : Spec(B) → Spec(A) is the morphism of schemes given by W ⊂ V . In the following, we view +every irreducible subset is in the corresponding affine schemes. We fix a pair (Q, �Q). Let f ∗[Q] = +m +� +i=1 +[Q′ +i], f ∗[ �Q] = +� +m +� +j=1 +[ �Q′ +j], [Q] · [Q] = +k� +p=1 +e(V, Q · �Q, Tp)[Tp] with T1 = T , and f ∗[Tp] = +lq� +q=1 +[T ′ +pq] with +T ′ +11 = T ′. Notice that each coefficient of [Q′ +i] in f ∗[Q] is 1 by Lemma 4.6, similar for f ∗[ �Q] and +f ∗[Tp]. We have +f ∗[Q] · f ∗[ �Q] = f ∗([Q] · [ �Q]), +20 + +i.e. +� +i,j +[Qi] · [ �Qj] = +� +i,j,p,q +e(W, Qi · �Qj, Tpq)[Tpq] = +� +p,q +e(V, Q, �Q, Tp)[Tpq], +where e(W, Qi · �Qj, Tpq) = 0 if Tpq ̸∈ Irr(Qi ∩ �Qj). Comparing the coefficient of [T11], we have +e(V, Q · �Q, T ) = � +i,j +e(W, Qi · �Qj, T ′). When (Q, �Q) runs through Irr(Y ∩ V ) × Irr(�Y ∩ V ) such that +T ∈ Irr(Q ∩ �Q), we have the equality we want. +Definition 5.5. Keep the notion in Lemma 5.4. We define the intersection product of Y and +�Y as +Y · �Y = +� +Z +eZ[Z] ∈ Zs+t(X), +where Z runs through the set Irr(Y ∩ �Y ) with codim(Z, X) = s + t, and eZ = e(X, Y · �Y , Z). +In general, let α = � +i∈I +ni[Yi] ∈ Zs(X) and β = � +j∈J +mj[�Yj] ∈ Zr(X). Assume that α and β +intersect properly. We define +α · β := +� +i,j +nimjYi · �Yj. +From the associativity of intersections in scheme theory, we have the associativity for our +definition. +Corollary 5.6. Keep the notion in Lemma 5.4. +Let Y, �Y , ��Y be irreducible Zariski-closed sub- +spaces of X. Assume that Y, �Y , ��Y intersect properly pairwise and that codim(Y ∩ �Y ∩ ��Y , X) = +codim(Y, X) + codim(�Y , X) + codim(��Y , X). Then +Y · (�Y · ��Y ) = (Y · �Y ) · ��Y +as cycles on X. +Proof. This is from Lemma 4.5 and the corresponding algebraic result, see [14, Lemma 43.20.1]. +Lemma 5.7. Let f : X → Y be flat morphism of regular K-analytic spaces. Let F, G be co- +herent sheaves on Y with codim(Supp(F), X) ≤ r, codim(Supp(G), X) ≤ s, and codim(Supp(F) ∩ +Supp(G), X) ≥ r+s+dim(Y )−dim(X). In this case, the cycle [f ∗F]r and [f ∗G]s intersect properly +and +f ∗([F]r · [G]s) = [f ∗F]r · [f ∗G]s. +Proof. This is from Lemma 4.5 and [14, Lemma 43.21.1] for regular, catenary Noetherian schemes. +The lemma implies the following corollary directly. +Corollary 5.8. Let f : X → Y be flat morphism of regular K-analytic spaces. Let α ∈ Zr(Y ), β ∈ +Zs(Y ). Assume that α and β intersect properly. Then f ∗α and f ∗β intersect properly and f ∗(α · +β) = f ∗α · f ∗β. +5.3 +Intersection multiplicities using Tor formula +We could define the multiplicities following the idea in [14, Section 43] by using TorOX +i +(F, G). +Firstly, it is not hard to see that TorOX +i +(F, G) is a coherent sheaf on X. Indeed, if X = M(A) +is affinoid, then Coh(X) ≃ Coh(Spec(A)). Since A is Noetherian, so we see that TorOX +i +(F, G) is a +coherent sheaf on X. For general case, +We show the following results. +Proposition 5.9. Let X be a regular, strictly K-analytic space. +21 + +(1) Let Y, �Y be irreducible Zariski-closed subspaces of X with codim(Y, X) = s, codim(�Y , X) = t. +Assume that Y, �Y intersect properly. Then +Y · �Y = +� +i +(−1)i[TorOX +i +(OY , O�Y )]s+t. +(2) Let F, G be coherent sheaves on X with codim(F, X) ≥ s, codim(F, X) ≥ t. Assume that +[F]s, [G]t intersecting properly. Then +[F]s · [G]t = +� +i +(−1)i[TorOX +i +(F, G)]s+t. +Proof. Obviously, (2) implies (1). By Lemma 4.5, Lemma 5.3 and Lemma 5.7, we can assume +that X is strictly affinoid. +Then this is [14, Lemma 43.19.4] for regular, catenary Noetherian +schemes. +6 +Projection formula +For a K-analytic spaces X, we denote D(Coh(X)) the derived category of Coh(X). We have the +derived tensor product ⊗L in D(Coh(X)), see [14, Definition 20.26.14]. If f : Y → X is a morphism +of K-analytic spaces, then we have a left derived functor +Lf ∗ : D(Coh(X)) → D(Coh(Y )) +see [14, Section 21.18]. If f is proper, we have a right derived functor +Rf∗ : D(Coh(Y )) → D(Coh(X)), +see [14, Section 21.19]. By adjointness of (Lf ∗, Rf∗), we have a morphism +Rf∗(E) ⊗L +OX F → Rf∗(E ⊗L +OY Lf ∗F), +see [14, Section 21.50]. As [14, Lemma 36.22.1], we have a similar result for K-analytic spaces. +Lemma 6.1. Let f : Y → X be a proper morphism of strictly K-analytic spaces. Then for any F +in D(Coh(X)) and E in D(Coh(Y )), the canonical morphism +Rf∗(E) ⊗L +OX F → Rf∗(E ⊗L +OY Lf ∗F) +is an isomorphism. +Proof. The proof is similar with the proof of [14, Lemma 36.22.1]. We can assume that X = M(A) +is affinoid. In this case, D(Coh(Y )) is the derived category of finitely generated A-modules, which +is a subcategory of D(A), the derived category of A-modules. We fix a coherent sheaf E on Y . For +an object M in D(A), we say that T (M) holds if the morphism +Rf∗(E) ⊗L +OX � +M → Rf∗(E ⊗L +OY Lf ∗ � +M) +is an isomorphism, where � +M is the corresponding sheaf of M on X. +If M = � +i +Mi and T (Mi) holds, then so does T (M). +Let N → L → M → N[1] be a +distinguished triangle in D(A). If T holds for two of N, L, M, then it holds for the third. Also +T (A[n]) for any shifts of A in D(A). +Hence T (M) holds for any object M in D(A), see [14, +Remark 15.59.11]. +Theorem 6.2 (Projection formula). Let f : Y → X be a flat, proper morphism of regular, +separated, strictly K-analytic spaces. Let α ∈ Z∗(Y ) and β ∈ Z∗(X). Assume that α and f ∗β +intersect properly. Then f∗(α) and β intersect properly and +f∗(α) · β = f∗(α · f ∗β). +22 + +Proof. Our proof is an analytic version of the proof of [14, Lemma 43.22.1] +By Lemma 5.3, Corollary 5.8 and Lemma 4.5, we can assume that X = M(A) is affinoid and +integral. Moreover, we assume α = [Z], β = [W] for some closed subspaces of dimension r and s. +If dimK f(Z) ̸= dimK Z, then f∗[Z] = 0, so f∗[Z] and [W] intersect properly. It sufficient to +show that f∗([Z] · f ∗[W]) = 0. We consider the morphism Z → f(Z), where f(Z) is endowed +with the reduced subspace structure. By Lemma 4.24, every fiber of Z → f(Z) has dimension +≥ 1. This implies that every fiber of the morphism Z ∩ f −1(W) → f(Z) ∩ W has dimension ≥ 1, +and dimK(Z ∩ f −1(W)) > dimK(f(Z) ∩ W). Since every irreducible component T of Z ∩ f −1(W) +has dimension dimK(Z ∩ f −1(W)), we conclude that dimK T > dimK f(T ). This implies what we +want. +If dimK f(Z) = dimK Z = r, then Z → f(Z) is finite. Let T ⊂ f(Z)∩W, and Ti ⊂ Z∩f −1(W), +i = 1, · · · , t be the irreducible components of Z ∩ f −1(W) dominating T . Since Z ∩ f −1(W) → +f(Z) ∩ W is finite, f is flat and Z, f −1(W) intersect properly, so +dimK T = dimK Ti = dimK Y − (dimK Y − r + dimK X − s) = r + s − dimK X, +Then f(Z) and W intersect properly. To show the equality, we follow the same idea of the proof +of [14, Lemma 42.23.1]. Since f is flat, by Lemma 6.1, we have +Rf∗(OZ) ⊗L +OX OW ≃ Rf∗(OZ ⊗L +OY f ∗OW ). +So for any generic point ξ ∈ Spec(A) corresponding to an irreducible component of f(Z) ∩ W, we +have +(f∗TorOY +i +(OZ, f ∗OW ))ξ = (TorOX +i +(f∗OZ, OW ))ξ. +(1) +On the other hand, by Proposition 5.9 and Proposition 4.29, we have +f∗([Z] · f ∗[W]) = +� +i +(−1)if∗[TorOY +i +(OZ, f ∗OW )]r+s−dimK Y += +� +i +(−1)i[f∗TorOY +i +(OZ, f ∗OW )]r+s−dimK Y , +f∗[Z] · [W] = [f∗OZ] · [W] += +� +i +(−1)i[TorOX +i +(f∗OZ, OW )]r+s−dimK Y . +Then f∗([Z] · f ∗[W]) = f∗[Z] · [W] by Eq. (1). +7 +GAGA +It is natural to expect that our definitions of cycles, flat pull-backs, proper push-forwards and +intersection products, for algebraic variety will be coincide with the ones in the intersection theory +of algebraic varieties. +Proposition 7.1. Let X be an algebraic variety over K. Then we have an isomorphism Z∗(X) ≃ +Z∗(Xan), +[Y ] �→ [Y an]. For a cycle α ∈ Z∗(X), we will denote its image in Z∗(Xan) by αan. +Moreover, the following properties hold. +(1) For any affinoid domain V contained in some affine open subset of Xan, the diagram diagram +commutes: +Z∗(X) +� +� Z∗(V) +� +Z∗(Xan) +� Z∗(V ) +, +where V = Spec(OXan(V )). +23 + +(2) Let α, β ∈ Z∗(X). Then α = β ∈ Z∗(X) (or αan = βan ∈ Z∗(Xan)) if and only if i∗α = +i∗β ∈ Z∗(V) for any any affinoid domain V contained in some affine open subset of Xan, +where V = Spec(OXan(V )) and i : V → X is the canonical morphism. +Proof. The map is obviously injective. It is suffices to show that every integral closed subspace +Z of Xan is algebraic. If X is proper over K, by GAGA result, see [2, Proposition 3.4.11], we +know that Z is algebraic. In general case, by Nagata’s compactification theorem, there is a proper +variety X over K such that X ⊂ X is an open immersion. We take the Zariski-closure Z of Z in +X +an, which is algebraic, i.e. there is an integral subvariety T ⊂ X such that T an = Z. We claim +that (T ∩ X)an = Z. By construction of analytification, we have (T ∩ X)an = T an ∩ Xan. We also +have Z ∩ Xan = Z. Then T an = Z implies that (T ∩ X)an = Z. +(1) The diagram is directly from the definition of [Y an] and Remark 4.8 (1). +(2) This is from the isomorphism Z∗(X) ≃ Z∗(Xan), the commutative diagram in (1) and +Lemma 4.5. +Remark 7.2. +(1) We have a surjection CH∗(X) ։ A∗(Xan). +Proposition 7.3. Let f : Y → X be a morphism of algebraic varieties over K. We have the +following hold. +(1) Let F be a coherent sheaf on X. Then [F]an = [Fan]. +(2) We have a canonical homomorphism Div(X) → Div(Xan), +D �→ Dan such that for any +D ∈ Div(X), we have [D]an = [Dan]. +(3) If ϕ is flat and α ∈ Z∗(X), then (ϕ∗(α))an = (ϕan)∗(αan). +(4) If ϕ is proper and β ∈ Z∗(Y ), then (ϕ∗(β))an = (ϕan)∗(βan). +(5) Let α, β ∈ Z∗(X). Then α, β intersect properly if and only if αan, βan ∈ Z∗(Xan) intersect +properly, and in this case, we have (α · β)an = αan · βan. +Proof. (1) Let V = M(B) ⊂ Xan be an affinoid domain contained in some affine open subsets of +Xan. Then we have a canonical morphism ϕ : Spec(A) → X which is flat by [7, TH´EOR`EM 3.3]. +It is sufficient to show that [F]an|V = [Fan]|V . +By the commutative diagram in (1), we have +[F]an|V = [ϕ∗F]; by Remark 4.8 (1), we have [Fan]|V = [Fan|V ] = [ϕ∗F]. So our claim holds. +(2) The homomorphism is given by the fact that V → X is flat for any an affinoid domain +V = M(A) ⊂ Xan contained in some affine open subsets of Xan, where V = Spec(A). Then the +compatibleness on such affinoid domains will induce a divisor on X. The equality can be proved +as (1). +(3) We take any affinoid domains V = M(A) ⊂ Xan and W = M(B) ⊂ Y an such that +ϕan(W) ⊂ V and V , W are contained in some affine open subsets of Xan, Y an respectively. Let +V = Spec(A), W = Spec(B). We have the following commutative diagram +W +j +� +�ϕ +� V +i +� +Y +ϕ +� X +Then +(ϕ∗(α))an|W = j∗ϕ∗(α) = �ϕ∗i∗(α) = (ϕan|W )∗(αan|V ) = (ϕan)∗(αan)|W . +here we identify the canonical isomorphisms Z∗(V ) ≃ Z∗(V) and Z∗(W) ≃ Z∗(W). By Lemma 4.5, +(3) follows. +(4) Since ϕ is proper, we have ϕan is proper. +We may assume that β is prime, moreover, +assume that X, Y are integral and β = [X], ϕ is finite, surjective. Hence we can assume that +X = Spec(A) and Y = Spec(B) are affine. Let V = M(A′) ⊂ Xan be an affinoid domain, and +24 + +U = (ϕan)−1(V ) = M(A′ ⊗A B). Notice that Frac(B) = B ⊗A Frac(A). We consider the following +diagram +Frac(A) ⊗A A′ +� Frac(B) ⊗A A′ +Frac(A) +� +� +Frac(B) +� +. +Notice that Frac(A) → Frac(B) is finite, so Frac(A) ⊗A A′ → Frac(B) ⊗A A′ is finite and flat. We +have that +[Frac(B) : Frac(A)] = +� +q,ϕ(q)=p +[(Frac(B) ⊗A A′)q : (Frac(A) ⊗A A′)p] +where q runs through the minimal ideal of Frac(B)⊗A A′, and we view ϕ : Spec(Frac(B)⊗A A′) → +Spec(Frac(A) ⊗A A′). The right-handed side is exactly deg(Y an/Xan) defined in Lemma 4.23, so +(4) holds. +(5) We can assume that α, β are prime. +Since flat pull-backs preserve proper intersection, +by Lemma 5.3, we know that α, β intersect properly if and only if αan, βan ∈ Z∗(Xan) intersect +properly. The proof of the equality is similar with the proof of (3). +8 +The category of finite correspondences +In this section, we will define the additive category CorK of finite correspondences of K-analytic +spaces. We will follow the notation in [1] and the idea in [13, Lecture 1]. +For the K-analytic spaces in this section, we always mean separated, quasi-paracompact, +strictly K-analytic spaces, the category of such spaces is exactly the category of separated, quasi- +paracompact, K-rigid spaces by [3, Theorem 1.6.1]. +A K-analytic space is said to be quasi-smooth if it is geometrically regular at each point, see +[8, Corollary 5.3.5]. In particular, a quasi-smooth space is regular. +Definition 8.1. Let X be a quasi-smooth, connected K-analytic space, and Y any K-analytic +space. An elementary correspondence from X to Y is an irreducible closed subset W of X ×Y +whose associated integral subspace is finite and surjective over X. +By an elementary corresponding from a quasi-smooth non-connected K-analytic space X to Y , +we mean an elementary correspondence from a connected component of X to Y . +The group CorK(X, Y ) is the free abelian group generated by the elementary correspondences +from X to Y . The element of CorK(X, Y ) are called finite correspondences. +Remark 8.2. +(1) If X is quasi-smooth, K-analytic space, one important example of elementary +correspondence from X to Y is the graph Γf of a morphism f : X → Y . If X is not connected, +the Γf is a finite correspondence from X to Y . Notice that Γf is closed in X × Y since Y is +separated and Γf is a section of X × Y → X. +(2) If X is not connected and X = � Xi is the decomposition into its connected components, we +have CorK(X, Y ) = � +i +CorK(Xi, Y ). +(3) Every closed subspace Z of X × Y which is finite and surjective over X determines a finite +correspondence [Z] from X to Y . +Proof. We only consider the case where X is connected. We can write [Z] = � +i +ni[Zi], where +Zi are irreducible component of Z such that Zi → X is surjective, and ni is the geometric +multiplicity of Zi of Z. +To define the composition of morphism in the category CorK, we need the following lemmas. +Lemma 8.3. Let f : T → T ′ be a morphism of K-analytic spaces over another K-analytic space +S. Let W be an irreducible Zariski-closed subset of T which is finite and surjective over S. Then +f(W) is irreducible, Zariski-closed in T ′ and finite, surjective over S. +25 + +Proof. Since T ′ → S is separated, W → S is finite, hence proper by [2, Corollary 3.3.8], we know +that W → T ′ is proper, see [5, Proposition 9.6.4]. So f(X) is irreducible Zariski-closed in T ′. +We replace T, T ′ by W, f(W) respectively, so we assume that T is finite and surjective over +S, and surjective on T ′. By [2, Corollary 3.3.8], it remains to show that T ′ is proper over S. +Obviously T ′ → S is quasi-compact since T → T ′ is surjective and T ′ → S quasi-compact. By [2, +Proposition 2.5.8 (iii)], we have +T = Int(T/S) = Int(T/T ′) ∩ f −1(Int(T ′/S)) = f −1(Int(T ′/S)), +this implies that Int(T ′/S) = T ′, i.e. ∂(T ′/S) = ∅. So T ′ is proper over S. +Lemma 8.4. Let Z be an integral K-analytic space, finite and surjective over a normal K-analytic +space S. Then for every morphism S′ → S with S′ connected (resp. irreducible), every connected +(resp. irreducible) component of Z ×S S′ is finite and surjective over S′. +Proof. This is in fact an algebraic result from [15, Proposition 2.17]. We can assume that S = +M(A), Z = M(B) and S′ = M(A′) are affinoid. Since B is finite over A, so B′ := B �⊗AA′ = +B ⊗A A′. +By [15, Proposition 2.17 (3)], we know that Spec(B) → Spec(A) is universally equidimensional, +hence universally open. +Then Spec(B′) → Spec(A′) is open. +For every connected component +T = M(C) of M(B′), the morphism Spec(C) → Spec(B′) is open. So M(C) → M(B′) has image +that is closed and Zariski-open, which is exactly M(B′) since it is connected. +For the irreducible case, since Spec(B′) → Spec(A′) is equidimensional. Then the image of each +irreducible component Spec(C) of Spec(B′) is Spec(A′). Since the image of M(C) is a Zariski- +closed subspace of M(A), it must be M(A). +Lemma 8.5. Let X, Y, Z be K-analytic spaces. Let V ⊂ X × Y and W ⊂ Y × Z be integral closed +subspace which are finite and surjective over X and Y respectively. Assume that Y is normal. +Then V × Z and X × W intersect properly in X × Y × Z, and each component of the push-forward +of the cycle [V × Z] · [X × W] on X × Z is finite and surjective over X. +Proof. Notice that V ×Y W ֒→ X × Y ×Y Y × Z ≃ X × Y × Z is the intersection of V × Z and +X × W in X × Y × Z, see the explanation in the remark. Then we have the following diagram +V ×Y W +� +� +W +� +� +Z +V +� +� +Y +X +. +By Lemma 8.4, each component of V ×Y W is finite and surjective over V , so it is also finite and +surjective over X, and it is of dimension dim X. This implies that V × Z and X × W intersect +properly in X × Y × Z. By Lemma 8.3, the image of each component of V ×Y W in X × Z is finite +and surjective over X. +Definition 8.6. Let CorK be the category defined as follows: +• Objects: the quasi-smooth K-analytic spaces; +• Morphisms: the finite correspondences CorK(X, Y ). +Given V ∈ CorK(X, Y ), W ∈ CorK(Y, Z), we define W ◦V as the push-forward of [V ×Z]·[X ×W] +on X × Z, which is an element in CorK(X, Z). +Remark 8.7. +(1) The composition is associative and bilinear, and the diagonal ∆X is the iden- +tity for a quasi-smooth K-analytic space X. +Proof. This is from Proposition 4.28 and Theorem 6.2, see the proof of [9, Proposition 16.1.1] +for the details. +26 + +(2) It is not hard to show that the category QSmK of quasi-smooth K-analytic spaces is fully +faithful subcategory of CorK. +(3) By [1, Proposition 2.2.35] and a few work, we can see our definition of CorK coincide with +[1, Definition 2.2.29]. +Following the idea in [4], we can define higher Chow groups CHn(X, s) for quasi-smooth K- +analytic spaces. +By GAGA principle, such definition will coincide with the one for algebraic +varieties. On the other hand, the higher Chow groups is also defined in [1, Introduction g´en´erale] +using motives of analytic spaces. It is natural to expect there is a close connection between these +two and higher Chow groups have similar properties as in the case of algebraic varieties. +Acknowledgements +The author would like to thank my host professor, Yigeng Zhao for his encouragement, support +and valuable suggestions. He would also like to thank Antoine Ducros, Walter Gubler and Michael +Temkin for their patience and answering questions during his study of Berkovich spaces. This +research is supported by postdoctoral research grant. +References +[1] Ayoub, J. (2015). Motifs des vari´et´es analytiques rigides. M´em. Soc. Math. Fr. (N.S.), (140- +141):vi+386. +[2] Berkovich, V. G. (1990). Spectral theory and analytic geometry over non-Archimedean fields, +volume 33 of Mathematical Surveys and Monographs. American Mathematical Society, Provi- +dence, RI. +[3] Berkovich, V. G. (1993). ´Etale cohomology for non-Archimedean analytic spaces. Inst. Hautes +´Etudes Sci. Publ. Math., (78):5–161 (1994). +[4] Bloch, S. (1986). Algebraic cycles and higher K-theory. Adv. in Math., 61(3):267–304. +[5] Bosch, S., G¨untzer, U., and Remmert, R. (1984). +Non-Archimedean analysis, volume 261 +of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical +Sciences]. Springer-Verlag, Berlin. A systematic approach to rigid analytic geometry. +[6] Ducros, A. (2007). Variation de la dimension relative en g´eom´etrie analytique p-adique. Compos. +Math., 143(6):1511–1532. +[7] Ducros, A. (2009). Les espaces de Berkovich sont excellents. Ann. Inst. Fourier (Grenoble), +59(4):1443–1552. +[8] Ducros, A. (2018). Families of Berkovich spaces. Ast´erisque, (400):vii+262. +[9] Fulton, W. (1998). +Intersection theory, volume 2 of Ergebnisse der Mathematik und ihrer +Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics +and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, +Berlin, second edition. +[10] Grothendieck, A. (1967). ´El´ements de g´eom´etrie alg´ebrique. IV. ´Etude locale des sch´emas et +des morphismes de sch´emas IV. Inst. Hautes ´Etudes Sci. Publ. Math., (32):361. +[11] Gubler, W. (1998). Local heights of subvarieties over non-Archimedean fields. J. Reine Angew. +Math., 498:61–113. +[12] Liu, Q. (2002). Algebraic geometry and arithmetic curves, volume 6 of Oxford Graduate Texts +in Mathematics. Oxford University Press, Oxford. Translated from the French by Reinie Ern´e, +Oxford Science Publications. +27 + +[13] Mazza, C., Voevodsky, V., and Weibel, C. (2006). +Lecture notes on motivic cohomology, +volume 2 of Clay Mathematics Monographs. American Mathematical Society, Providence, RI; +Clay Mathematics Institute, Cambridge, MA. +[14] Stacks project authors, T. (2022). The stacks project. https://stacks.math.columbia.edu. +[15] Voevodsky, V., Suslin, A., and Friedlander, E. M. (2000). +Cycles, transfers, and motivic +homology theories, volume 143 of Annals of Mathematics Studies. Princeton University Press, +Princeton, NJ. +Y. Cai, Westlake University, Dunyu Road 600, Xihu District 310024, Hangzhou, China +E-mail address: caiyulin@westlake.edu.cn +28 + diff --git a/7tE0T4oBgHgl3EQfwQFU/content/tmp_files/load_file.txt b/7tE0T4oBgHgl3EQfwQFU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5d61e8aecca4a8a2b56047ddbcbe23c1e175874d --- /dev/null +++ b/7tE0T4oBgHgl3EQfwQFU/content/tmp_files/load_file.txt @@ -0,0 +1,1319 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf,len=1318 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='02629v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='AG] 31 Oct 2022 Intersection theory on non-archimedean analytic spaces Yulin Cai January 9, 2023 Abstract We develop the intersection theory of non-archimedean analytic spaces and prove the pro- jection formula and the GAGA principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' As an application, we naturally define the category of finite correspondences of analytic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Contents 1 Introduction 1 2 Preliminary 3 3 Meromorphic functions and Cartier divisors 7 4 Cycles, flat pull-backs and proper push-forwards 11 5 Proper intersection and intersection multiplicities 19 6 Projection formula 22 7 GAGA 23 8 The category of finite correspondences 25 Acknowledgements 27 References 27 1 Introduction The intersection theory of non-archimedean analytic spaces has been studied in [11, Section 2] and [1, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='2], and the author believes that some experts have concrete idea about such a theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In [11], Gubler considers the Cartier divisors on rigid analytic spaces and formal schemes, and define their intersection with irreducible analytic subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This theory allows him to define the local height of subvarieties over non-archimedean fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In [1], Ayoub develops the theory of motives on rigid analytic spaces using homotopy theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' He uses the presheaves on the category of affinoid spaces to construct the category of finite corre- spondence (for rigid analytic space) RigCor(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Such construction avoids the intersection theory of analytic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In this paper, we will develop the intersection theory of non-archimedean analytic spaces follow- ing the idea similar to the case of algebraic varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We will show the flat base change formula, the projection formula and the GAGA principle to relate the intersection theories of analytic spaces and of algebraic varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' As an application, we will give a direct construction of RigCor(K) (simply denoted by CorK in this paper) like [13, Lecture 1] does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In fact, we can define the higher Chow groups of analytic spaces as [4] for algebraic varieties, and this definition is different from Ayoub’s in [1, Introduction g´en´erale].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In Section 2, we give some basic notion in the theory of Berkovich spaces, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' support of a coherent sheaf, Zariski image and codimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We also extend [7, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='12] into an abstract form, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='15 which is a key lemma for this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' With this lemma, we can solve the compatibility problems in our theory, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='6 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 1 In Section 3, we define and study the Cartier divisors on an analytic space X, which form a group Div(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The group of divisors up to linear equivalence is denoted by CaCl(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' As in the theory of schemes, we have an injective homomorphism CaCl(X) ֒→ Pic(X), and it is an isomorphism if X is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In Section 4, we give the notion of cycles, and associate a coherent sheaf with a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In particular, we can associate a closed subspace with a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' As in the theory of algebraic varieties, the flat pull-backs and proper push-forwards of cycles are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We prove the following flat base change formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='1 (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let Y ′ g′ � f ′ � Y f � X′ g � X be a Cartesian diagram of separated, strictly K-analytic spaces with f proper and g flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then f ′ is proper, g′ is flat and g∗ ◦ f∗ = f ′ ∗ ◦ g′∗ on Z∗(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In Section 5, we define intersection product of proper intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We will give two definitions, meaning a local one using the scheme theory and a global using Tor formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For a flat morphism f : Y → X of K-analytic spaces of pure dimension, the pull-back f ∗ : Z∗(X) → Z∗(Y ) preserves intersection product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Since we have the flat pull-backs, proper push-forwards and intersection products, the expected projection formula is proved in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='2 (Projection formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let f : Y → X be a flat, proper morphism of regular, separated, strictly K-analytic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let α ∈ Z∗(Y ) and β ∈ Z∗(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Assume that α and f ∗β intersect properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then f∗(α) and β intersect properly and f∗(α) · β = f∗(α · f ∗β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In Section 7, we compare the intersection theories of algebraic varieties and of non-archimedean analytic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We prove the GAGA principle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In Section 8, we define the category of finite correspondence CorK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This category is also defined by Ayoub [1] using another definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Notation and terminology Throughout this paper, we fix a complete non-archimedean field K with a non-trivial valuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For a K-analytic space, we mean a Berkovich space over K, see [3, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The structure sheaf on a K-analytic space X with respect to the G-topology is denoted by OX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If it is necessary, we will use the notation XG for the G-topology instead of the ordinary topology on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The (K- analytic) dimension of X is denoted by dimK X, or dim X when there is no confusion with the fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Given a point x ∈ X, H (x) denotes its complete residue field and dimx X denotes the local dimension of X at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We shall simply say ”coherent sheaf on X” for ”coherent OX-module (with respect to G- topology)”, and denote Pic(X) for the group of invertible sheaves on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Assume that X is good, let F be a coherent sheaf on X and x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We denote by Fx the stalk at x of F viewed as a sheaf of the underlying ordinary topology of X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Fx := lim −→ U F(U) = lim −→ V F(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' where U runs through open neighborhoods of x, and V runs through affinoid neighborhoods of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We will write Irr(X) for the set of all irreducible components of X, and write Irr(X) for the set of all irreducible Zariski-closed subsets of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Notice that Irr(X) has a partial order: W ≤ Z if W ⊂ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 2 For an algebraic variety over K, we mean a separated scheme of finite type over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For a commutative ring A, R(A) denotes the set of all regular elements of A and Frac(A) = R(A)−1A, the maximal localization containing A as a subring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 2 Preliminary For the convenience of the reader and further uses, in the section, we provide some basic concepts and results that are either given somewhere, or formulated easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='1 Support of a coherent sheaf (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' [8, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='5]) Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let X be a K-analytic space, F be a coherent sheaf on X, and Ann(F) be the (coherent) annihilator ideal of F (on the site XG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The support of F is the closed analytic subspace of X defined by Ann(F), denoted by Supp(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) Recall the annihilator I of F is defined as follows: for any analytic domain V , Ann(F)(V ) := {a ∈ OX(V ) | a · F(V ) = 0}, which is a coherent ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In particular, for any analytic domain V , we have Ann(F)|V = Ann(F|V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (2) If X = M(A) is affinoid and F = � M for some finitely generated A-module, then it is easy to see that Ann(F) = � Ann(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' From the definition, we can easy deduce the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let X be a K-analytic space, F a coherent sheaf on X, and Z = Supp(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then there is a unique coherent sheaf G on Z such that F = i∗G, where i : Z ֒→ X is the canonical immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By uniqueness, we can glue coherent sheaf G from local parts, so we can assume that X = M(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' It is not hard to see the lemma in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='2 Zariski image of a morphism As in the theory of schemes, we can define Zariski image of a morphism of analytic spaces, which has a natural structure of analytic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We follow the idea in [14, Subsection 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let X be a K-analytic space, F a coherent sheaf on X, and G ⊂ F an OX-submodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then there is a unique coherent OX-submodule G′ ⊂ G with the following property: for any coherent OX-module H, the canonical map HomOX(H, G′) → HomOX(H, G) is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In particular, G′ is the largest coherent sheaf contained in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let {Gi}i∈I be the set of coherent sheaves contained in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We consider the morphism of OX-modules ϕ : � i∈I Gi → F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We claim its image G′ ⊂ G is coherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let pG′ ⊂ G be the image of ϕ as presheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then G′ is the sheafification of pG′, and for any affinoid domain V = M(V ), pG′(V ) = � i Gi(V ) ⊂ F(V ) is a finitely generated A-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By Tate acyclic theorem, we have G′(V ) = pG′(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' So G′ is coherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' It is the largest coherent sheaf contained in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 3 The map HomOX(H, G′) → HomOX(H, G) is obviously injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For any homomorphism ψ : H → G ⊂ F, the image Im(ψ) ⊂ G is a coherent sheaf, so Im(ψ) ⊂ G′, so f factor thorough G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This implies that G′ is the one we want.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For the uniqueness, if G′′ is another coherent OX-submodule with the universal property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then the bijectivity of HomOX(G′, G′′) → HomOX(G′, G) implies that we have a homomorphism G′ → G′′ ⊂ G, so G′ ⊂ G′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Hence G′ = G′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let f : Y → X be a morphism of K-analytic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then there is a closed analytic subspace Z of X such that (a) the morphism f factors through Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (b) (Universal property) if f factors through a closed analytic subspace Z′ of X, then Z′ contains Z as a closed analytic subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The closed analytic space Z of X is called the Zariski image of f, denoted by Imzar(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By (b), if Z exists, then it is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' It remains to show the existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let I := Ker(OY → f∗OX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='4, we take the largest coherent OX-submodule J ⊂ I and set Z = V (J ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' It remains to check (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (a) We have f(Y ) ⊂ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Indeed, for any affinoid domain V = M(A) ⊂ X and any affinoid domain U = M(B) ⊂ f −1(V ), we have J (V ) ⊂ I(V ) ⊂ Ker(A → B), so U → V factors through M(A/J (V )) = Z ∩V and f(U) ⊂ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Hence f(Y ) ⊂ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We denote the map Y → Z by f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We shall construct f # : OZ(V ∩ Z) → OX(f −1(V )) for any affinoid domain V ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Since J (V ) ⊂ I(V ), the homomorphism OX(V ) → OY (f −1(V )) factor through OZ(V ∩Z) = OX(V )/J (V ), we denote OZ(V ∩Z) → OX(f −1(V )) by f # which is compatible on intersections of affinoid domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Hence we have a morphism f : Y → Z and f = i ◦ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (b) If f factors through a closed subspace Z′ of X with Z′ = V (J ′), then J ′ ⊂ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By the choice of J , we have J ′ ⊂ J , so Z′ ⊂ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) Locally, f : M(B) → M(A) is given by ϕ : A → B, then Imzar(f) = M(A/ Ker(ϕ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We may expect the Zariski image is exactly the usual image as sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' It is almost true if Y is reduced or f is quasi-compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let f : Y → X be a morphism of K-analytic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If Y is reduced, then the Imzar(f) = f(Y ) Xzar with the reduce closed subspace structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' As a map, f factor through f(Y ) Xzar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Since Y is reduced, so f factors through f(Y ) Xzar with the reduced structure, see [7, PROPOSITION 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='2 (iii)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' It remains to show the universal property of Y → f(Y ) Xzar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If f factors through a closed subspace Z of X, then f(Y ) Xzar ⊂ Z as a subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The containment is also a morphism of analytic spaces since f(Y ) Xzar is endowed with the reduced structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let f : Y → X be a morphism of K-analytic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Assume that f is quasi-compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) I = Ker(OX → f∗OY ) is coherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In particular, Imzar(f) = V (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (2) f(X) Xzar = Imzar(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In other word, Y → Imzar(f) is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (3) For any analytic domain V ⊂ X, the subspace Imzar(f)∩V is the Zariski image of f|f −1(V ) : f −1(V ) → V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 4 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) Suppose X = M(A) is affinoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We take a G-covering Y = n� i=1 Vi by affinoid domains, and set Y ′ = n� i=1 Vi, π : Y ′ → Y the canonical morphism which is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For any analytic domain V ⊂ Y , the map π# : OY (V ) → OY ′(π−1(V )) = n � i=1 OY (V ∩ Vi) is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We consider f ′ := f ◦ π : Y ′ → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then I = Ker(OX → f ′ ∗OY ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Since Y ′ is affinoid, so I = (Ker(A → OY ′(Y ′))∼ which is coherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This implies (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (3) This is from (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (2) By (3), suffices to assume that X = M(A) is affinoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We use the notations in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Notice that f(Y ) Xzar = f ′(Y ′) Xzar, so we can assume that Y = M(B) is affinoid, and f is induced by ϕ : A → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We have I = � Ker(ϕ) and Imzar(f) = M(A/ Ker(ϕ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' So the morphism X → Imzar(f) is induced by an injective homomorphism A/ Ker(ϕ) → B, hence it is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='3 Codimension We recall the definition of codimension in [8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let X be a K-analytic space, and Y a Zariski-closed subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The codimen- sion codim(Y, X) of Y in X is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If both Y and X are irreducible, codim(Y, X) := dimK X − dimK Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If Y is irreducible, codim(Y, X) := sup Z∈Irr(X) Y ⊂Z codim(Y, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In the general case, codim(Y, X) := inf Z∈Irr(Y ) codim(Z, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For x ∈ X, we define the codimension of Y in X at x as codimx(Y, X) := \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 inf Z∈Irr(Y ) x∈Z codim(Z, X) if x ∈ Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' +∞ if x ̸∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) Let W ⊂ Z ⊂ Y ⊂ X be irreducible closed analytic subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then codim(W, Y ) = codim(W, Z) + codim(Z, Y ), dimK(Z) + codim(Z, Y ) = dimK(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='11 ([6] Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let X = M(A) be a K-affinoid space, Y = V (I) for some ideal I ⊂ A, and x ∈ X with image ξ ∈ Spec(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then (1) codim(Y, X) = codim(Spec(A/I), Spec(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (2) codimx(Y, X) = codimξ(Spec(A/I), Spec(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) In particular, (1) implies that codim(Spec(AL/IL), Spec(AL)) = codim(Spec(A/I), Spec(A)) for any complete field extension L/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Or we can write dimK X − dimK Y = codimKrull(Y, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 5 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let X be a K-analytic space, and Z, Y ∈ Irr(X) with Z ⊂ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then codim(Z, Y ) = max{m | Z = Y0 ⊊ Y1 ⊊ · · · ⊊ Ym = Y }, where Yi ∈ Irr(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Moreover, each maximal chain has the same length, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' every K-analytic space is catenary with respect to the Zariski topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Firstly, if Z ⊊ Y , then codim(Z, Y ) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This can be seen locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Hence ”≥” holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Conversely, it suffices to show that if codim(Z, Y ) ≥ 2, then there is W ∈ Irr(X) such that Z ⊊ W ⊊ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Indeed, we take an affinoid domain V of Y are affinoid, and V = M(A), Z∩V = M(A/I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then we know that codim(Z, Y ) = codim(Spec(A/I), Spec(A)) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' So we can find a prime ideal p ∈ Spec(A) such that W := M(A/p) Yzar strictly contains Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Apply the same method, we can see that each maximal chain has the same length (this in fact due to the additivity of codimension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) In particular, we see that the codimension is independent of the base field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='4 A key lemma For a set S satisfying certain conditions, we can determine if S satisfies a property P or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In this case, we say that the property P is well-defined on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' It is not well-defined if S does not satisfy these conditions at the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The following generalized result from [7, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='12] is crucial for extending a local result on irreducible closed subsets to be global.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let X be a K-analytic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let P be a property on irreducible components satisfying the following properties: there is a G-covering X = � i∈I Vi by affinoid domain, the property P is well-defined (this means that we can determine if P is satisfied or not) on each irreducible component of Vi (or simply say that P is well-defined on Vi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' if P is well-defined on an irreducible component Z of an affinoid domain V , then P is well- defined on each irreducible component of W for any affinoid domain W ⊂ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Moreover, in this case, for any irreducible component T of W ∩ Z, we have T satisfies P ⇐⇒ Z satisfies P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then there exist Zariski-closed subsets X+ P , X− P of X which are characterized by the following properties: for any affinoid domain V on which P is well-defined, we have X+ P ∩ V = � T ∈Irr(V ), T satisfies P T, X− P ∩ V = � T ∈Irr(V ), T doesn’t satisfy P T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Notice that X = X+ P ∪ X− P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For any affinoid domain V on which P is well-defined, set C+(V ) := {T ∈ Irr(V ) | T satisfies P}, C−(V ) := {T ∈ Irr(V ) | T doesn’t satisfy P}, E+(V ) := � T ∈C+(V ) T, E−(V ) := � T ∈C−(V ) T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 6 Let V be an affinoid domain on which P is well-defined, and W ⊂ V an affinoid domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let Z be an irreducible component of V and T an irreducible component of W containing Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By our assumption, T ∈ C+(W)⇐⇒ Z ∈ C+(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By [7, COROLLAIRE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='11], we have E+(W) = E+(V ) ∩ W and E−(W) = E−(V ) ∩ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let X+ P (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' X− P ) be the union of E+(V ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' E−(V )) where V is an affinoid domain on which P is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then for any affinoid domain V of X on which P is well-defined, we have X+ P ∩ V = E+(V ) and X− P ∩ V = E−(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Since P is well-defined on Vi for some G-covering X = � i∈I Vi by affinoid domain, and E+(Vi), E−(Vi) ⊂ Vi are Zariski-closed, so X+ P , X− P ⊂ X are Zariski-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 3 Meromorphic functions and Cartier divisors The sheaf of meromorphic functions and Cartier divisors are defined on a ringed space in [10, Section 20, Section 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' On a G-ringed space, these definitions do not work since the restriction of a regular element is not necessarily regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Fortunately, this can be remedied on analytic spaces (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' [11, Section 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In this section and next section, we will following the idea in [10, Section 20, Section 21] to discuss meromorphic functions, Cartier divisors and cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='1 Meromorphic functions For a (commutative) ring A, denote R(A) ⊂ A the set of all regular elements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' non-zero divisors, we know R(A) is a multiplicative set, and the corresponding localization Frac(A) := R(A)−1A is the maximal localization containing A as a subring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let X be a K-analytic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For any affionid domain V = M(A) ⊂ X, we set K′ X(V ) := Frac(A), this will defined a presheaf on affinoid domains on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The associated sheaf KX with respect to the G-topology on X is called the sheaf of meromorphic functions on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' An element of KX(X) is called a meromorphic function on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The subsheaf of invertible elements of KX is denoted by K∗ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) For affinoid domains U = M(B) ⊂ V = M(A) of X, and f ∈ R(A), the restriction of f on U is in R(B), this implies that our definition of KX is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' It is from the fact A → B is flat, or we assume that B = A{p−1 1 T1,··· ,p−1 n Tn} (gT1−f1,··· ,gTn−fn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (2) For any analytic domain V ⊂ X, we have KX(V ) = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 (si)i ∈ � i K′ X(Vi) �������� V = � i Vi is a G-covering of V with Vi affi- noid and si|Vijk = sj|Vijk for some G-covering Vi ∩ Vj = � k Vijk with Vijk affinoid \uf8fc \uf8f4 \uf8f4 \uf8fd \uf8f4 \uf8f4 \uf8fe � ∼, where (si)i ∼ (s′ j)j if for any i, j, there exists a G-covering Vi ∩V ′ j = � k Vijk with Vijk affinoid such that si|Vijk = s′ j|Vijk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If X is separated, then it can be simplified as KX(V ) = � (si)i ∈ � i K′ X(Vi) ����� V = � i Vi is an G-covering of V with Vi affi- noid and si|Vi∩Vj = sj|Vi∩Vj � � ∼, where (si)i ∼ (s′ j)j if for any i, j, si|Vi∩V ′ j = s′ j|Vi∩V ′ j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (3) For any affinoid domain V ⊂ X, the canonical map K′ X(V ) → KX(V ) is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In particular, OX ⊂ KX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Given an affinoid domain V and any finite G-covering V = n� i=1 Vi by affinoid domains, let A = OX(V ) and Ai = OX(Vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We consider the restriction map Frac(A) → n� i=1 Frac(Ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let a/b ∈ Frac(A) be such that its restriction on Frac(Ai) is 0 for any i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' a = 0 ∈ Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This implies that a = 0 ∈ A by Tate’s acyclic theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Hence K′ X(V ) ֒→ KX(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We take a G-covering X = � i∈I Vi by affinoid domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then the injective map OX(Vi) ֒→ K′ X(Vi) will induce OX ֒→ KX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Keep the notion in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For an OX-module F, we call F ⊗OX KX the sheaf of meromorphic sections of F, and we have a canonical map idF ⊗i : F → F ⊗OX KX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The sheaf F is called strictly without torsion if idF ⊗i is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' A global section of F ⊗OX KX is called a meromorphic sections of F on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If F is coherent on X, we say a meromorphic section s on X is defined on a Zariski-open subset V if s|V is in the image of F(V ) via idF ⊗i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If moreover, F is strictly without torsion, then there is a maximal Zariski-open subset V on which s is defined, such V is called the domain of definition of s, denoted by dom(s) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' s ∈ F(dom(s))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) Notice that F → F ⊗OX KX is the sheafification of the presheaf given by V �→ F(V ) ⊗OX(V ) K′ X(V ) for any affinoid domain V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' So for any analytic domain V ⊂ X, we have (F ⊗OX KX)|V ≃ F|V ⊗OV KV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In particular, KX|V = KV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (2) A locally free OXG-module F is strictly without torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Moreover, F ⊗OX KX is a KX- module, here, we view (XG, KX) as a G-ringed space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For a good, strictly K-analytic space, the sheaf of meromorphic functions can be given in a similar way in [10, Section 20], and will have some good properties, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' properties for schemes can be extended to good analytic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If X is good, strictly K-analytic, and x ∈ X is rigid, we have that OX,x = lim −→ V OX(V ) where V runs through affinoid domains containing x, see [2, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In particular, it suffices that V runs through (strictly) affinoid neighborhoods of x in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let X be a good, strictly K-analytic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For any analytic domain V ⊂ X, set R(V ) := {s ∈ OX(V ) | sx ∈ R(OX,x) for any x ∈ V } ⊂ OX(V ), which defines a sheaf on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then the following statements hold: (1) For any affinoid domain V ⊂ X, we have R(V ) = R(OX(V )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In particular, and KX to be the sheafification of the following presheaf: for any analytic domain V ⊂ X, V �→ R(V )−1OX(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (2) For any rigid point x ∈ X, we have K′ X,x ≃ Frac(OX,x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For any analytic domain V ⊂ X, the canonical homomorphism K′ X(V ) ֒→ � x∈V rigid K′ X,x is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 8 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Notice that the presheaf R is a sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Since R is a subpresheaf of OX, and if V = � i∈I Vi is a G-covering of an analytic domain V , ai ∈ R(Vi) such that ai|Vi∩Vj = aj|Vi∩Vj then there exists a ∈ OX(V ) such that a|Vi = ai, then a ∈ R(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) For any affinoid domain V ⊂ X and a ∈ OX(V ), we have a is regular ⇐⇒ a ∈ OX,x regular for any x ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Indeed, ”=⇒” is from the flatness, for ”⇐=”, if a ∈ OX,x is regular, then there is an affinoid neighborhood Vx of x in V such that a ∈ R(OX(Vx)) (since Ker(OX(V ) a → OX(V )) is finitely generated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then a ∈ R(OX(V )) since V = � x∈V Vx is a G-covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' So R(V ) = R(OX(V )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Hence K′ X(V ) = Frac(OX(V )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (2) By definition, we have a map lim −→ V K′ X(V ) → R−1 x OX,x which is surjective, where V runs through affinoid neighborhoods of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If a/b ∈ K′ X(V ) with V affinoid neighborhood of x such that a/b = 0 ∈ R−1 x OX,x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' there is c ∈ Rx such that ac = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We can assume that c ∈ OX(V ), then a/b = 0 ∈ K′ X(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' It remains to show that Rx = R(OX,x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We have an injective map Rx ֒→ R(OX,x) by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Conversely, for a ∈ R(OX,x), we consider an affinoid neighborhood V of x with A = OX(V ) such that a ∈ A, then 0 � Ann(a) � A � A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Since Ann(a) is finitely generated and a ∈ R(OX,x), so we can find an affinoid neighborhood U ⊂ V of x with B = OX(U) such that Ann(a) ⊗A B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' So a ∈ R(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By (1), we know that Rx = R(OX,x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If a/b ∈ K′ X(V ) such that 0 = a/b ∈ K′ X,x for any rigid x ∈ V , then there exists an affinoid neighborhood Vx of x such that 0 = a/b ∈ K′ X(Vx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Since R(Vx) = R(OX(Vx)), we have 0 = a ∈ OX(Vx) and a = 0 ∈ K′ X(V ), a/b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='2 Cartier divisors Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let K be a complete non-archimedean field, and X a K-analytic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We denote the group H0(XG, K∗ X/O∗ X) by Div(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The elements of Div(X) are called Cartier divisors of XG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let f ∈ H0(XG, K∗ X), its image in Div(X) is called a principal Cartier divisor and denoted by div(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We say that two Cartier divisor D1, D2 are linearly equivalent if D1 − D2 is principal, write D1 ∼ D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We denote CaCl(X) the group of equivalent class of Cartier divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' A Cartier divisor D is called effective if it is in the image of the canonical map H0(XG, (OX ∩ K∗ X)/O∗ X) → H0(XG, K∗ X/O∗ X), write D ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The set of effective Cartier divisors is denoted by Div+(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) The exact sequence of sheaves 0 � O∗ X � K∗ X � K∗ X/O∗ X � 0 will induce a long exact sequence 0 � H0(XG, O∗ X) � H0(XG, K∗ X) � Div(X) � H1(XG, O∗ X) � H1(XG, K∗ X) � · · · (2) We can represent a Cartier divisor D by a system {(Ui, fi)}i∈I, where X = � i∈I Ui is a G- covering by affinoid domains, and fi = ai/bi ∈ K′ X(Ui) such that fi|Ui∩Uj ∈ fj|Ui∩UjOX(Ui∩ Uj)∗ for every i, j ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Two systems {(Ui, fi)}i∈I and {(Vj, gj)}j∈J represent the same Cartier divisor if only only if fi|Ui∩Vj ∈ gj|Ui∩VjOX(Ui ∩ Vj)∗ for any i ∈ I, j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 9 If D1 = {(Ui, fi)}i∈I and D2 = {(Vj, gj)}j∈J, then D1 + D2 = {(Wijk, figj)}i∈I,j∈J, where Ui ∩ Vj = � k Wijk is a G-covering by affinoid domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In particular, if X = M(A) is affinoid, let X = Spec(A), then we have an injection Div(X) ֒→ Div(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Keep the notion in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) For any divisor D = {(Ui, fi)}i∈I ∈ CaCl(X), we can associate a subsheaf OX(D) ⊂ KX defined by OX(D)|Ui = f −1 i OX|Ui, which is an invertible sheaf and independent of the choice of representative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Moreover, D ≥ 0 ⇐⇒ OX(−D) ⊂ OX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (2) The construction above gives a homomorphism of groups ρ : Div(X) → Pic(X), D �→ OX(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (3) The homomorphism ρ induces an injective homomorphism CaCl(X) → Pic(X) with image Im ρ = {L ∈ Pic(X) | L ֒→ KX}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (4) If X is affinoid and reduced, then ρ : CaCl(X) → Pic(X) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We follow the idea of the proof of [12, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) Assume D = {(Vj, gj)}j∈J is another representative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then OX(D)|Ui∩Vj = f −1 i OX|Ui∩Vj = (gju)−1OX|Ui∩Vj = g−1 j OX|Ui∩Vj where u ∈ OX(Ui ∩ Vj)∗, this implies OX(D) is independent of the choice of representative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By construction, OX(D) ∈ Pic(X), and D ≥ 0 if and only if OX(D) ⊂ OX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (2) The map is a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Indeed, let D1 = {(fi, Ui)}i∈I and D2 = {(gi, Ui)}i∈I, then ρ(D1 + D2)|Ui = f −1 i g−1 i OX|Ui ≃ f −1 i OX|Ui ⊗OX|Ui g−1 i OX|Ui, and this isomorphism is compatible on the intersection Ui ∩ Uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (3) If D = {(Ui, fi)}i∈I = div(f) is a principal divisor with f ∈ H0(XG, K∗ X) and fi = f|Ui ∈ K′ X(Ui), where X = � i∈I Ui is a G-covering of X by affinoid domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then f −1 ∈ OX(D)(X) because of the following exact sequence 0 � OX(D)(X) � � i∈I f −1 i OX(Ui) � � i∈I f −1 i OX(Ui ∩ Uj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' So we can define the morphism OX → OX(D), a �→ af −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' It is an isomorphism since it is an isomorphism on each Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Hence we have a homomorphism CaCl(X) → Pic(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If D = {(Ui, fi)}i∈I ∈ Div(X) such that OX(D) ≃ OX, then there is g ∈ OX(D)(X) such that the morphism OX ∼ → OX(D), a �→ ag is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Since OX(D)|Ui ≃ f −1 i OX|Ui = g|UiOX|Ui and f −1 i ∈ K′∗ X(Ui), g|Ui = f −1 i ui ∈ K′∗ X(Ui) ⊂ K∗ X(Ui) with ui ∈ O∗ X(Ui), we have g ∈ K∗ X(X) and D = {(Ui, fi)}i∈I = {(Ui, g−1|Ui)}i∈I is principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By definition, we know that OX(D) ⊂ KX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Conversely, for L ∈ Pic(X) with L ⊂ KX, there is G-covering X = � i∈I Ui by affinoid domains such that OX|Ui ≃ L|Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We take gi ∈ L(Ui) which is mapped to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then gi ∈ KX(Ui) and L|Ui = giOX|Ui, moreover, there is fi ∈ K∗ X(Ui) such that figi = 1 because of the isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' On Ui ∩ Uj, we have L|Ui∩Uj = f −1 i OX|Ui∩Uj = f −1 j OX|Ui∩Uj, so there is u ∈ O∗ X(Ui ∩ Uj) such that f −1 i |Ui∩Uj = uf −1 j |Ui∩Uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then L = OX(D), where D = {(Ui, fi)}i∈I ∈ Div(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 10 (4) Let X = Spec(OX(X)), then CaCl(X) ≃ Pic(X), see [12, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We a commuta- tive diagram Div(X)� � � ∼ � Div(X) ρ � Pic(X) ∼ � Pic(X) , so our claim holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The isomorphism Pic(X) ≃ Pic(X) is from Coh(X) ≃ Coh(X) and Tate’s acyclic theorem, see the proof of [3, Propostion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='4 (iii)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) We know that H1(XG, O∗ X) ≃ Pic(X), then ρ is the connecting map of the long exact sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let L be a line bundle on a normal K-analytic space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let s ∈ H0(X, L⊗OX KX) be a rational section which is non-zero on each irreducible component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let X = � i∈I Ui be a G- covering of X by integral affinoid domains such that L|Ui is free and generated by an element ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then these exist fi ∈ K∗ X(Ui) such that s|Vi = fiei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Moreover div(s) := {(Ui, fi)}i∈I is a Cartier divisor such that OX(div(s)) ≃ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='3 Inverse image of a Cartier divisor Next we consider the restriction of Cartier divisors on a closed analytic subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let D ∈ Div(X), and Z ∈ Irr(X) with reduced analytic space structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='We say D intersects Z properly if there is a G-covering X = � i∈I Ui by affinoid domains such that D = {(Ui, ai/bi)}i∈I with the images ai, bi ∈ R(OZ(Ui∩Z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The set of Cartier divisor intersecting Z properly is a subgroup of Div(X), denoted by GZ/X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) There is a natural homomorphism GZ/X → Div(Z) denoted by D �→ D|Z, compatible with the homomorphism OX → i∗OZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Moreover, we have a canonical isomor- phism OX(D)|Z ≃ OZ(D|Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 4 Cycles, flat pull-backs and proper push-forwards 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='1 Cycles Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let X be a K-analytic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' A prime cycle on X is an element in Irr(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' A cycle on X is a formal sum α = � Z∈Irr(X) nZ[Z] with nZ ∈ Z which is G-locally finite, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' the set {Z ∈ Irr(X) | Z ∩ V ̸= ∅, nZ ̸= 0} is finite for any affinoid domain V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The coefficient nZ is called the multiplicity of α at Z, denoted by multZ(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We say that a cycle α is positive if multZ(α) ≥ 0 for any Z ∈ Irr(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The set of cycles (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' positive cycles) is denoted by Z(X) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Z+(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The union of the Z such that nZ ̸= 0 is called the support of α, denoted by Supp(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' It is a Zariski-closed subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By convention, Supp(0) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' A cycle α is (purely) of codimension r (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' of dimension r) if any Z ∈ Irr(X) with nZ ̸= 0 has codimension r (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' dimension r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The cycles of codimension r (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' of dimension r) form a subgroup Zr(X) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Zr(X)) of the group of cycles on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) For a positive cycle α = � Z∈Irr(X) nZ[Z] and any Z ∈ Irr(X) with nZ ≥ 1, we can endow Z with the reduced subscheme structure, then Z = V (IZ) is an integral closed analytic subspace of X, where IZ is the coherent sheaf of ideal defining Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We view α as a closed analytic subspace defined by the sheaf of ideal Iα := � Z∈Irr(X) InZ Z and we have a 11 canonical closed immersion j : α = V (Iα) ֒→ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This induces a homomorphisms of semi- groups Z+(X) → {closed analytic subspace of X} = {coherent sheaves of ideals on X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (2) By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='13, we know that Zr(X) is not dependent of the base field K, but Zr(X) is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let X = M(A) be a K-affinoid space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Set X = Spec(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then Div(X) ֒→ Div(X), Z∗(X) ≃ Z∗(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The first arrow is also an isomorphism if X is regular, see Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let X be a K-analytic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let α ∈ Z+(X) with associated sheaf of ideal Iα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then V (Iα) = Supp(α) with Irr(V (Iα)) = {maximal elements in α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This is local, and we can deduce this lemma from the example above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The following lemma is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let X = � i∈I V be a G-covering of by affinoid domains, and α, β ∈ Div(X) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Z∗(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then α = β ⇐⇒ α|Vi = β|Vi for any i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' It suffices to show the ”if” part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If α, β ∈ Div(X), then the result holds from the expression of Cartier divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If α = � Z nZ[Z], β = � Z mZ[Z] ∈ Zk(X) such that α|Vi = β|Vi for any i ∈ I, then nZ[Z ∩ Vi] = mZ[Z ∩ Vi] for any Z ∈ Irr(X) with Z ∩ Vi ̸= ∅, so nZ = mZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='2 Cycle associated to a coherent sheaf We will construct a homomorphism Div(X) → Z1(X) as we do in algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Recall, for a Noetherian affine scheme X = Spec(A), a coherent sheaf F = � M on X, and an irreducible component Z of Supp(F), we set multZ(F) := lengthAp(Mp), called the multiplicity of Z in F, where p ∈ X is the prime ideal corresponding to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For a divisor D ∈ Div(X) and a codimension one prime cycle Z = {z} ∈ Z1(X), we set multZ(D) := multOX,z(Dz) the multiplicity of Z in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For an affinoid space M(A), we have similar notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let X be a K-analytic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let F be a coherent sheaf on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For any irreducible component Z of Supp(F) with reduced analytic space structure, and an affinoid domain V ⊂ X with Z ∩ V ̸= ∅, we set multZ(F) := multT (F|V ) where T is an irreducible component of Z ∩ V with T Supp(F)Zar = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then multZ(F) is a positive integer which is independent of the choice of T and V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We call multZ(F) the multiplicity of Z in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For a fixed irreducible component Z of Supp(F), and any affinoid domain V, W ⊂ X with W ⊂ V , Z ∩ W ̸= ∅, we claim that multT (F|V ) = multT ′(F|W ) where T ∈ Irr(Z ∩ V ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='T ′ ∈ Irr(Z ∩ W)) with T ′VZar = T , T XZar = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Indeed, let V = M(A), W = M(B) and F|V = � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We shall show that lengthAp(Mp) = lengthBq(Mp ⊗Ap Bq) where p ⊂ A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' q ⊂ B) is the prime ideal corresponding to T (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' T ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Notice that the kernel W → Spec(B) is surjective, we can find a y ∈ W such that Ker(| · |x) = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let x ∈ V be the image of y, then Ker(| · |x) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We have H (x) = H (y) and lengthAp(Mp) = dimk(p)(M ⊗A k(p)) = dimH (x)(M ⊗A H (x)), 12 it is similar for lengthBq(Mp ⊗Ap Bq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Hence our claim holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' To show the lemma, we apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let Z ∈ Z1(X) be a prime cycle, and m = multT (F|V ) for some affinoid domain V ⊂ X with Z ∩ V ̸= ∅, where T ∈ Irr(Z ∩ V ) with T XZar = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For V given as before, we say an irreducible component T ∈ Irr(Z ∩ V ) satisfies P if multT (F|V ) = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' After replacing X by Z, from our claim, we see that P satisfies the hypothesis in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then there are Zariski-closed subsets Z+ P , Z− P of Z such that Z+ P ∩ V = � T ∈Irr(Z∩V ), T satisfies P T, Z− P ∩ V = � T ∈Irr(Z∩V ), T doesn’t satisfy P T, and Z = Z+ P ∪ Z− P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Since Z is irreducible and there is some T ⊂ Z+ P , we have Z = Z+ P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This implies the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Keep the notion in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For a coherent sheaf F on X with codim(Supp(F), X) ≥ k, we set [F]k := � Z∈Irr(Supp(F))k multZ(F)[Z] ∈ Zk(X), called the cycle associated to F with codimension k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='6, it is hard to have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let V = M(A) ⊂ X is an affinoid domain, and F a coherent sheaf on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Set V = Spec(A) and Fal V the coherent sheaf on V corresponding to F|V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then [F|V ]k = [Fal V ]k, here we identify Z∗(V ) ≃ Z∗(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Keep the notion in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For a closed analytic subspace Y of X with codim(Y, X) ≥ k, we set multZ(Y ) := multZ(OY ), for any Z ∈ Irr(Y ), called the multiplicity of Z in Y , and set [Y ]k := � Z∈Irr(Y ) Z∈Zk(X) multZ(Y )[Z] ∈ Zk(X), called the cycle associated to Y with codimension k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='3 Weil divisors Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let X be a K-analytic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' An element in Z1(X) is called a Weil divisor on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let X be a K-analytic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let D ∈ Div(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For any prime cycle Z ∈ Z1(X), and any affinoid domain V ⊂ X with Z ∩ V ̸= ∅, D|V ∈ K′ X(V ), we set multZ(D) := multT (D|V ) where T ∈ Irr(Z ∩ V ) with T XZar = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then multZ(D) is independent of the choice of T and V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We call multZ(D) the multiplicity of Z for D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The proof is similar with the one of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For any prime cycle Z ∈ Z1(X) and any affinoid domain V, W ⊂ X with W ⊂ V , Z ∩ W ̸= ∅, D|V ∈ K′ X(V ), we claim that multT (D|V ) = multT ′(D|W ), where T ∈ Irr(Z ∩V ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='T ′ ∈ Irr(Z ∩W)) with T ′VZar = T , T XZar = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Indeed, since both sides are additive, we can assume that D|V = f ∈ R(OX(V )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let Y ⊂ V be a closed analytic subspace determined by f ∈ OX(V ), then our claim is from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' To show the lemma, we apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let m = multT (D|V ) for some affinoid domain V ⊂ X with Z ∩ V ̸= ∅, D|V ∈ K′ X(V ), where T ∈ Irr(Z ∩ V ) with T XZar = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For V given as before, we say an irreducible component T ∈ Irr(Z ∩ V ) satisfies P if multT (D|V ) = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' After replacing X by Z, from our claim, we see that P satisfies the hypothesis in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then there are Zariski-closed subset Z+ P , Z− P of Z such that Z+ P ∩ V = � T ∈Irr(Z∩V ), T satisfies P T, Z− P ∩ V = � T ∈Irr(Z∩V ), T doesn’t satisfy P T, and Z = Z+ P ∪ Z− P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Since Z is irreducible, and there is some T ⊂ Z+ P , so Z = Z+ P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This implies the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let X be a K-analytic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For any D ∈ Div(X), we set [D] := � Z⊂Irr(X) codim(Z,X)=1 multZ(D)[Z] ∈ Z1(X), called the Weil divisor associated to D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In particular, for any f ∈ K∗(X), we denote (f) := [div(f)] ∈ Z1(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Such a divisor (f) is called a principal divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The set of principal divisors Rat1(X) form a subgroup of Z1(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We denote the quotient of Z1(X) by the subgroup of principal divisors by Cl(X) := Z1(X)/Rat1(X), called the class group of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We say that two divisors Z, Z′ are rationally equivalent and write Z ∼rat Z′ if they have the same class in Cl(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Recall, a K-analytic space X is regular at x ∈ X if there is a good analytic domain V of X containing x such that OV,x is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We say X is regular if X is regular at every point x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This is equivalent to that for any affinoid domain V ≃ M(A) ⊂ X, we have that A is regular, see [8, Lemma-Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='1, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The map [·] : Div(X) → Z1(X) a homomorphism which sends effective divisors to positive cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This induces a homomorphism [·] : CaCl(X) → Cl(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If X is normal (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' regular), then these two map are injective (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' isomorphic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' It is easy to see that [·] : Div(X) → Z1(X) is a homomorphism and induces [·] : CaCl(X) → Cl(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If X is normal, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='5, to show [·] : Div(X) → Z1(X) is injective, we can assume X is affinoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For D ∈ Div(X) such that multZ(D) = 0 for any Z ∈ Z1(X), we take affinoid domain V ⊂ X with Z ∩ V ̸= ∅ and D|V ∈ K′ X(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then D|V ∈ O∗ X(V ) since multT (D|V ) = 0 for any Q ∈ Z1(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This implies that D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' As for the quotient, if [D] = (f) for some f ∈ K∗ X(X), then D = div(f), this implies that [·] : CaCl(X) → Cl(X) is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Assume that X is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' To show that [·] : Div(X) → Z1(X) is surjective, we firstly assume that X = M(A) is affinoid and set X = Spec(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In this case, Div(X) ≃ Z1(X), see [12, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Hence, we have a commutative diagram Div(X)� � � ∼ � Div(X) ρ � Z1(X) ∼ � Z1(X) , 14 so our claim holds for affinoid spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We can glue Cartier divisors on affinoid domains together by injectivity of [·].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Hence [·] : Div(X) → Z1(X) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='4 Rational equivalence of cycles As in the classical definition of Chow group of an algebraic variety, we can extend the class group for any codimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let X be a K-analytic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For any (k + 1)-dimensional irreducible closed analytic subspace W of X and any f ∈ K∗ W (W), we have a k-cycle [div(f)] ∈ Zk(W) ⊂ Zk(X) given in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' A k-cycle α is rationally equivalent to zero, write α ∼ 0, if there are a finite number of (k + 1)-dimensional subvarieties Wi of X, and fi ∈ K∗ Wi(Wi) such that α = � i [div(fi)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Since [div(f −1)] = −[div(f)], the cycles rationally equivalent to zero form a subgroup Ratk(X) ⊂ Zk(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The group of k-cycles modulo rational equivalence on X is the quotient Ak(X) := Zk(X)/Ratk(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Define Z∗(X) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' A∗(X)) to be the direct sum of the Zk(X) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Ak(X)) for k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' A cycle class on X is an element of A∗(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' A cycle class is positive if it can be represented by a positive cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) The subgroup Ratk(X) ⊂ Zk(X) is well-defined by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (2) Ak(X) = Ak(Xred) for any k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (3) If X is of pure dimension n, then An(X) = Zn(X) is the free abelian group generated by the irreducible components of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='5 Flat pull-backs We have introduced Cartier divisors, cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Next we consider their pull-backs via flat morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Recall the definition of flatness in sense of [8, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='8], a morphism f : Y → X of K- analytic spaces is naively flat if for any y ∈ Y , there exist a good analytic domain V ⊂ Y containing y and a good analytic domain U ⊂ X containing f(V ) such that OV,y is flat over OU,f(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We say f is flat if moreover Y ′ := Y ×X X′ → X′ is naively flat for any morphism X′ → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If f is flat, then OY (V ) is flat over OX(U) for any affinoid domains V ⊂ Y and U ⊂ X with f(V ) ⊂ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The converse is not true in general unless f is locally finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Notice that for any analytic domain V of X, the natural morphism V ֒→ X is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' A morphism f : Y → X of K-analytic spaces has relative dimension r if for any Z ∈ Irr(X), f −1(Z) = ∅ or any irreducible component Z′ of f −1(Z) has dimK Z′ = dimK Z+r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) The notion of relative dimension r is an analogue of the one in algebraic geometry, see [9, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Our definition is different from the one in [8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We don’t assume that such morphisms are surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let f : Y → X be a flat morphism of K-analytic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then f has relative dimension r if and only if Yx = ∅ or Yx is of equidimension r for any x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In particular, if f : Y → X is flat with X, Y equidimensional, then f has relative dimension dimK Y − dimK X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We apply [8, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='11] saying that dimy Y = dimy Yx + dimx X for any y ∈ Yx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Assume that f has relative dimension r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If x ∈ X such that Yx ̸= ∅, then for any Z ∈ Irr(X) containing x, we have dimK f −1(Z)−dimK Z = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This implies that dimy Yx = dimy Y −dimx X = r for any y ∈ Yx since dimx X = max x∈Z∈Irr(X){dimK Z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 15 Conversely, for any Z ∈ Irr(X) with f −1(Z) ̸= ∅, without loss of generality, we can assume that Z = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We take y ∈ Y and x = f(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then dimy Y = dimx X + dimy Yx = dimK X + r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This implies that f has relative dimension r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If X, Y are equidimensional, then dimy Yx = dimy Y − dimx X implies that Yx is of equidimen- sion for any y ∈ Y, x = f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let f : Y → X be a flat morphism of K-analytic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) The canonical morphism f # : OX → f∗OY extends to a morphism f # : K∗ X/O∗ X → f∗(K∗ Y /O∗ X), then we have a homomorphism f ∗ : Div(X) → Div(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This will induce a homomorphism f ∗ : CaCl(X) → CaCl(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (2) Assume that X, Y are of equidimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For any integral closed subspace Z ⊂ X of pure codimension k, we set f ∗[Z] := [f −1(Z)] ∈ Zk(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This extends by linearity to a pull-back homomorphism f ∗ : Zk(X) → Zk(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) The flat pull-backs are functorial and we have a commutative diagram Div(X) f ∗ � [·] � Div(Y ) [·] � Z1(X) f ∗ � Z1(Y ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let f : Y → X be a flat morphism of K-analytic spaces of pure dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For a coherent sheaf F on X with codim(Supp(F), X) ≥ k, we have codim(Supp(f ∗F), X) ≥ k and [f ∗F]k = f ∗[F]k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In particular, if Z is a closed analytic subspace of X of pure codimension k, then f ∗[Z] = [f −1(Z)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We can reduce the statement to the case of affinoid spaces by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='5, then the proposi- tion from the analogue result in scheme theory by Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='8 (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For the result in scheme theory, see proof of [14, Lemma 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='4 (2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='6 Proper push-forward of cycles For an affinoid space X = M(A), it may happen that dimKrull A < dimK X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In order to avoid this dimension problem, we assume that all K-analytic spaces (including affinoid domains) in this subsection are strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In this case dimKrull A = dimK X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Recall a theorem of Kiehl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='22 ([2] Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let f : Y → X be a proper morphism of K-analytic spaces, and F a coherent OY -module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then Rnf∗F, n ≥ 0, are coherent OX-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In particular, we have Remmert’s mapping theorem, saying that f(Y ) is an Zariski-closed subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' A similar result of the following lemma is given in [11, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let f : Y → X be a surjective finite morphism of integral, strictly K-analytic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For any (strictly) affinoid domain V ⊂ X and T ∈ Irr(V ), we set deg(Y/X) := � Q∈Irr(f −1(V )) f(Q)=T [Frac(AQ) : Frac(AT )], where AT , AQ are the affinoid algebras corresponding to T, Q with reduced structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then deg(Y/X) is independent of the choice of V and T , called the degree of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 16 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Apply the usual technique with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='15, it is sufficient to show that for any affinoid domain V, W ⊂ X with W ⊂ V , and any T ∈ Irr(V ), T ′ ∈ Irr(W), we have � Q∈Irr(f −1(V )) f(Q)=T [Frac(AQ) : Frac(AT )] = � Q′∈Irr(f −1(W)) f(Q′)=T ′ [Frac(AQ′) : Frac(AT ′)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This is in fact from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='6 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='21 for affinoid case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let V = M(A), f −1(V ) = M(B) and W = M(A′), then f −1(W) = M(B′), where B′ = A′⊗AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let F be the corresponding coherent sheaf associated to B as an A-module on V , and i : W → V the canonical morphism, then [F]0 = � T ∈Irr(V ) ( � Q∈Irr(f −1(V )) f(Q)=T [Frac(AQ) : Frac(AT )])[T ], and we know that � Q∈Irr(f −1(V )) f(Q)=T [Frac(AQ) : Frac(AT )] is independent of the choice of T by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We also have i∗[F]0 = � T ∈Irr(V ) ( � Q∈Irr(f −1(V )) f(Q)=T [Frac(AQ) : Frac(AT )]) � T ′∈Irr(T ∩W) [T ′], [i∗F]0 = � T ′∈Irr(W) ( � Q′∈Irr(f −1(W)) f(Q′)=T ′ [Frac(AQ′) : Frac(AT ′)])[T ′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='21, we compare the coefficient of some for any irreducible component T ′, we can see that our claim holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We have the following equivalent conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let f : Y → X be a morphism of integral, separated, strictly K-analytic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then the following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (i) f is surjective and finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (ii) f is surjective, proper, and dimK Y = dimK X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='a) f is proper, and for any x ∈ X, dimH (x) f −1(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='b) f is proper, and for any rigid point x ∈ X, f −1(x) ̸= ∅ has finite rigid points as an H (x)- analytic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='a) f is surjective and proper, and there is a point x ∈ X such that dimH (x) f −1(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='b) f is surjective and proper, and there is a rigid point x ∈ X such that dimH (x) f −1(x) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' f −1(x) ̸= ∅ and has finite rigid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Obviously, (i) =⇒ (iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='a), (iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='b) =⇒ (iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='b) =⇒ (iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='a) =⇒ (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This is from [8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='14 (3)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (ii) =⇒ (iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Since f is quasi-compact, after taking irreducible components of affinoid domain of X, Y , we can assume that X = M(A), Y = M(B) are affinoid, integral and dim A = dim B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Moreover, since the original morphism is surjective, we know that the corresponding morphism ϕ : Spec(B) → Spec(A) is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For any closed point x ∈ Spec(A) with ϕ−1(x) ̸= ∅, by basic property of strict affinoid algebras, we know that codim(x, Spec(A)) = dim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Since ϕ is dominant, then dim B ≥ codim(x, Spec(A)) + dim ϕ−1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' So dim ϕ−1(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Notice that K → A → H (x) is finite, then H (x) is the residue field of Spec(A) at x, and B ⊗A H (x) = B �⊗AH (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Hence the rigid points of f −1(x) is exactly the closed points of ϕ−1(x) which are finite since B �⊗AH (x) Noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 17 (iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='b) =⇒ (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The separatedness ensure that X, Y are also rigid K-analytic spaces, see [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then the result is from [5, Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='6] and [2, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='a) =⇒ (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Notice that we have proved the equivalence (i) ⇐⇒ (ii) ⇐⇒ (iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='a) ⇐⇒ (iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By [6, TH´EOR`EME 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='9], the set {y ∈ Y | dimy f ≥ 1} is Zariski-closed in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' So {x ∈ X | dimH (x) f −1(x) ≥ 1} = f({y ∈ Y | dimy f ≥ 1}) is Zariski-closed in X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' U := {x ∈ X | dimH (x) f −1(x) ≤ 0} is Zariski-open in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then dimK f −1(U) = dimK U by the equivalence of (iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='a) and (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Since dimK Y = dimK f −1(U), dimK X = dimK U, we have (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' With the lemmas above, we have the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let f : Y → X be a proper morphism of separated, strictly K-analytic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For any irreducible closed subspace Z of Y , the image f(Z) is a Zariski-closed subset of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We set deg(Z/f(Z)) := � the degree of f : Z → f(Z) if dimK f(Z) = dimK Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 0 if dimK f(Z) < dimK Z (notice that dimK f(Z) = dimK Z is equivalent to f : Z → f(Z) is finite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Define f∗[Z] := deg(Z/f(Z))[f(Z)], then extends linearly to a homomorphism (of gradding groups) f∗ : Z∗(Y ) → Z∗(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) For Z above, we know that f(Z) with the reduced subspace structure is the Zariski image of Z → X by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We can easily prove the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let f : Y → X and g : Z → Y be proper morphism of separated strictly K-analytic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then g∗ ◦ f∗ = (g ◦ f)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let Y ′ g′ � f ′ � Y f � X′ g � X be a Cartesian diagram of separated, strictly K-analytic spaces with f proper and g flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then f ′ is proper, g′ is flat and g∗ ◦ f∗ = f ′ ∗ ◦ g′∗ on Z∗(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The morphism f ′ is proper by [3], and g′ is flat by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For the equality, notice that it holds if f is a closed immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In general, To show g∗(f∗α) = f ′ ∗(g′∗(α)), we can assume that α = [Y ] and it is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Moreover, we can assume that X = f(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If dimK X < dimK Y , then left-handed side is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For any x′ ∈ X′, let x = g(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We have (f ′)−1(x′) = M(H (x′)) ×X′ Y ′ = M(H (x′)) ×X Y = M(H (x′)) ×H (x) f −1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Since f is not finite, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='24 (iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='a), we have dimH (x′)(f ′)−1(x) = dimH (x) f −1(x) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This means that f ′ is not finite, and f ′∗([Y ′]) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If dimK X = dimK Y , then f : Y → X is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='5, it suffices to consider the affine case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then the result is from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='21, and can be proved similarly as Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' With the proposition above, we can always assume that the base space is affinoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We can use this to deduce the following result to the scheme case, see [14, Lemma 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='4] for the scheme version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 18 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let f : Y → X be a proper morphism of separated strictly K-analytic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) Let Z ⊂ Y be a closed subspace with dimK Z ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then f∗[Z]k = [f∗OZ]k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (2) Let F be a coherent sheaf on X such that dimK(Supp(F)) ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then f∗[F]k = [f∗F]k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Obviously, it suffices to show (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='3, there is a coherent sheaf G on Z := Supp(F) such that F = i∗G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let Z′ be the Zariski image of Z → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Notice that f(Z) = Z′ by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='8 and properness of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' So we have the following commutative diagram Z� � � f|Z � Y f � Z′� � � X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By functorial property of push-forward, it suffices to show (f|Z)∗[G] = [(f|Z)∗G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' So we can assume that dimK X = k and f : X → Y is proper and dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Moreover, we can assume that Y is affinoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' So dimK Y ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We write f∗[F]k = � W nW [W] and [f∗F]k = � W mW [W] where W runs through irreducible component of X of dimension k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For a fixed irreducible com- ponent W, to show nW = mW , it suffices to show that (f∗[F]k)|V = ([f∗F]k)|V for some affinoid domain V ⊂ X with V ∩ W ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We can take Zariski-open subsets U ⊂ X such that U ∩ W ′ = ∅ and U ∩ f(T ) = for any irreducible component W ′ of X which is distinct from W, and any ir- reducible component T of Y which doesn’t dominate W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We can take an affinoid domain of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' So we can assume X = M(A) is equidimensional and each irreducible component of Y dominates some irreducible component of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By [2, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='8], we know that Y is finite over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' So we reduce to the case where Y, X is affinoid and f is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This is an algebraic result, see the last part of the proof of [14, Lemma 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 5 Proper intersection and intersection multiplicities 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='1 Proper intersection Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let X be a regular K-analytic space of pure dimension, and Y, �Y ∈ Irr(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then for every irreducible component Z of Y ∩ �Y , we have codim(Z, X) ≤ codim(Y, X) + codim(�Y , X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The proof is based on the corresponding result in scheme theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We can assume that X is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For any affinoid domain V ⊂ X, we have codim(T, V ) = codim(Y, X), where T is a irreducible component of V ∩Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then we can apply the corresponding result in scheme theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let X be a regular K-analytic space of pure dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) Let Y, �Y ∈ Irr(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We say that Y and �Y intersect properly if codim(Z, X) ≥ codim(Y, X)+ codim(�Y , X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (2) Let α = � i∈I ni[Yi] ∈ Zs(X) and β = � j∈J mj[�Yj] ∈ Zr(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We say that α and β intersect properly if Yi and �Yj intersect properly for all i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 19 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let X be a regular K-analytic space of pure dimension, and Y, �Y ∈ Irr(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then the following statements are equivalent: (i) Y, �Y intersect properly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (ii) For any x ∈ Y ∩ �Y , there is an affinoid domain V containing x such that any Q ∈ Irr(Y ∩ V ), �Q ∈ Irr(�Y ∩ V ) intersect properly on V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (iii) For any affinoid domain V with Y ∩ V , �Y ∩ V ̸= ∅ and any Q ∈ Irr(Y ∩ V), �Q ∈ Irr(�Y ∩ V ), we have Q and �Q intersect properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For any affinoid domain V ⊂ X with Y ∩ V = ∅ and any Q ∈ Irr(Y ∩ V ), we have codim(Q, V ) = codim(Y, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then the lemma follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='2 Multiplicities and intersect products In this subsection, we will apply the intersection theory on a regular catenary Noetherian scheme to define multiplicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Another definition using Tor formula will be given in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Recall, on a regular, catenary Noetherian scheme X, let Q, �Q be irreducible closed subschemes with codim(Q, X) = s, codim( �Q, X) = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then intersection product of Q, �Q is defined by Q · �Q = � T eT[T ] := � i (−1)i[TorOX i (OQ, O � Q)]s+t ∈ Zs+t(X), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' eT = e(X, Q · �Q, T ) = � i (−1)ilengthOX,T (TorOX,T i (OQ,T , O � Q,T )) where T runs through Irr(Q ∩ �Q) with codim(T, X) = s + t, and OX ,T (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' OQ,T , resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' O � Q,T ) denotes the local ring of X (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Q, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' �Q) at the generic point of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let X be a regular K-analytic space of pure dimension, and Y, �Y ⊂ X irreducible Zariski-closed subspaces with codim(Y, X) = s, codim(�Y , X) = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Assume that Y and �Y intersect properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For any irreducible component Z of Y ∩ �Y with codim(Z, X) = s + t, and any affinoid domain V ⊂ X with Z ∩ V ̸= ∅, we set e(X, Y · �Y , Z) := � Q, � Q e(V, Q · �Q, T ) where T ∈ Irr(Z ∩ V ) and (Q, �Q) runs through Irr(Y ∩ V ) × Irr(�Y ∩ V ) such that T ∈ Irr(Q ∩ �Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then e(X, Y, �Y , Z) is a positive integer which is independent of the choice of V and T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We call e(X, Y, �Y , Z) the multiplicity of Z on Y ∩ �Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The idea of proof is similar with the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='6 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' It is sufficient to show that for any affinoid domain V, W ⊂ X with W ⊂ V , Z ∩ W ̸= ∅, we have that � Q, � Q e(V, Q · �Q, T ) = � Q′, � Q′ e(W, Q′ · �Q′, T ′) where T ∈ Irr(Z ∩ V ), (Q, �Q) runs through Irr(Y ∩ V ) × Irr(�Y ∩ V ) such that T ∈ Irr(Q ∩ �Q), and T ′, Q′, �Q′ is given similarly with T ′VZar = T , T XZar = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let V = M(A), W = M(B) and f : Spec(B) → Spec(A) is the morphism of schemes given by W ⊂ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In the following, we view every irreducible subset is in the corresponding affine schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We fix a pair (Q, �Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let f ∗[Q] = m � i=1 [Q′ i], f ∗[ �Q] = � m � j=1 [ �Q′ j], [Q] · [Q] = k� p=1 e(V, Q · �Q, Tp)[Tp] with T1 = T , and f ∗[Tp] = lq� q=1 [T ′ pq] with T ′ 11 = T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Notice that each coefficient of [Q′ i] in f ∗[Q] is 1 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='6, similar for f ∗[ �Q] and f ∗[Tp].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We have f ∗[Q] · f ∗[ �Q] = f ∗([Q] · [ �Q]), 20 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' � i,j [Qi] · [ �Qj] = � i,j,p,q e(W, Qi · �Qj, Tpq)[Tpq] = � p,q e(V, Q, �Q, Tp)[Tpq], where e(W, Qi · �Qj, Tpq) = 0 if Tpq ̸∈ Irr(Qi ∩ �Qj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Comparing the coefficient of [T11], we have e(V, Q · �Q, T ) = � i,j e(W, Qi · �Qj, T ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' When (Q, �Q) runs through Irr(Y ∩ V ) × Irr(�Y ∩ V ) such that T ∈ Irr(Q ∩ �Q), we have the equality we want.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Keep the notion in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We define the intersection product of Y and �Y as Y · �Y = � Z eZ[Z] ∈ Zs+t(X), where Z runs through the set Irr(Y ∩ �Y ) with codim(Z, X) = s + t, and eZ = e(X, Y · �Y , Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In general, let α = � i∈I ni[Yi] ∈ Zs(X) and β = � j∈J mj[�Yj] ∈ Zr(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Assume that α and β intersect properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We define α · β := � i,j nimjYi · �Yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' From the associativity of intersections in scheme theory, we have the associativity for our definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Keep the notion in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let Y, �Y , ��Y be irreducible Zariski-closed sub- spaces of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Assume that Y, �Y , ��Y intersect properly pairwise and that codim(Y ∩ �Y ∩ ��Y , X) = codim(Y, X) + codim(�Y , X) + codim(��Y , X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then Y · (�Y · ��Y ) = (Y · �Y ) · ��Y as cycles on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This is from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='5 and the corresponding algebraic result, see [14, Lemma 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let f : X → Y be flat morphism of regular K-analytic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let F, G be co- herent sheaves on Y with codim(Supp(F), X) ≤ r, codim(Supp(G), X) ≤ s, and codim(Supp(F) ∩ Supp(G), X) ≥ r+s+dim(Y )−dim(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In this case, the cycle [f ∗F]r and [f ∗G]s intersect properly and f ∗([F]r · [G]s) = [f ∗F]r · [f ∗G]s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This is from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='5 and [14, Lemma 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='1] for regular, catenary Noetherian schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The lemma implies the following corollary directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let f : X → Y be flat morphism of regular K-analytic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let α ∈ Zr(Y ), β ∈ Zs(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Assume that α and β intersect properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then f ∗α and f ∗β intersect properly and f ∗(α · β) = f ∗α · f ∗β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='3 Intersection multiplicities using Tor formula We could define the multiplicities following the idea in [14, Section 43] by using TorOX i (F, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Firstly, it is not hard to see that TorOX i (F, G) is a coherent sheaf on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Indeed, if X = M(A) is affinoid, then Coh(X) ≃ Coh(Spec(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Since A is Noetherian, so we see that TorOX i (F, G) is a coherent sheaf on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For general case, We show the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let X be a regular, strictly K-analytic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 21 (1) Let Y, �Y be irreducible Zariski-closed subspaces of X with codim(Y, X) = s, codim(�Y , X) = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Assume that Y, �Y intersect properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then Y · �Y = � i (−1)i[TorOX i (OY , O�Y )]s+t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (2) Let F, G be coherent sheaves on X with codim(F, X) ≥ s, codim(F, X) ≥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Assume that [F]s, [G]t intersecting properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then [F]s · [G]t = � i (−1)i[TorOX i (F, G)]s+t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Obviously, (2) implies (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='5, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='3 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='7, we can assume that X is strictly affinoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then this is [14, Lemma 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='4] for regular, catenary Noetherian schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 6 Projection formula For a K-analytic spaces X, we denote D(Coh(X)) the derived category of Coh(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We have the derived tensor product ⊗L in D(Coh(X)), see [14, Definition 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If f : Y → X is a morphism of K-analytic spaces, then we have a left derived functor Lf ∗ : D(Coh(X)) → D(Coh(Y )) see [14, Section 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If f is proper, we have a right derived functor Rf∗ : D(Coh(Y )) → D(Coh(X)), see [14, Section 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By adjointness of (Lf ∗, Rf∗), we have a morphism Rf∗(E) ⊗L OX F → Rf∗(E ⊗L OY Lf ∗F), see [14, Section 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' As [14, Lemma 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='1], we have a similar result for K-analytic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let f : Y → X be a proper morphism of strictly K-analytic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then for any F in D(Coh(X)) and E in D(Coh(Y )), the canonical morphism Rf∗(E) ⊗L OX F → Rf∗(E ⊗L OY Lf ∗F) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The proof is similar with the proof of [14, Lemma 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We can assume that X = M(A) is affinoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In this case, D(Coh(Y )) is the derived category of finitely generated A-modules, which is a subcategory of D(A), the derived category of A-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We fix a coherent sheaf E on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For an object M in D(A), we say that T (M) holds if the morphism Rf∗(E) ⊗L OX � M → Rf∗(E ⊗L OY Lf ∗ � M) is an isomorphism, where � M is the corresponding sheaf of M on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If M = � i Mi and T (Mi) holds, then so does T (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let N → L → M → N[1] be a distinguished triangle in D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If T holds for two of N, L, M, then it holds for the third.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Also T (A[n]) for any shifts of A in D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Hence T (M) holds for any object M in D(A), see [14, Remark 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='2 (Projection formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let f : Y → X be a flat, proper morphism of regular, separated, strictly K-analytic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let α ∈ Z∗(Y ) and β ∈ Z∗(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Assume that α and f ∗β intersect properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then f∗(α) and β intersect properly and f∗(α) · β = f∗(α · f ∗β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 22 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Our proof is an analytic version of the proof of [14, Lemma 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='1] By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='3, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='8 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='5, we can assume that X = M(A) is affinoid and integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Moreover, we assume α = [Z], β = [W] for some closed subspaces of dimension r and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If dimK f(Z) ̸= dimK Z, then f∗[Z] = 0, so f∗[Z] and [W] intersect properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' It sufficient to show that f∗([Z] · f ∗[W]) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We consider the morphism Z → f(Z), where f(Z) is endowed with the reduced subspace structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='24, every fiber of Z → f(Z) has dimension ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This implies that every fiber of the morphism Z ∩ f −1(W) → f(Z) ∩ W has dimension ≥ 1, and dimK(Z ∩ f −1(W)) > dimK(f(Z) ∩ W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Since every irreducible component T of Z ∩ f −1(W) has dimension dimK(Z ∩ f −1(W)), we conclude that dimK T > dimK f(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This implies what we want.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If dimK f(Z) = dimK Z = r, then Z → f(Z) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let T ⊂ f(Z)∩W, and Ti ⊂ Z∩f −1(W), i = 1, · · · , t be the irreducible components of Z ∩ f −1(W) dominating T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Since Z ∩ f −1(W) → f(Z) ∩ W is finite, f is flat and Z, f −1(W) intersect properly, so dimK T = dimK Ti = dimK Y − (dimK Y − r + dimK X − s) = r + s − dimK X, Then f(Z) and W intersect properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' To show the equality, we follow the same idea of the proof of [14, Lemma 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Since f is flat, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='1, we have Rf∗(OZ) ⊗L OX OW ≃ Rf∗(OZ ⊗L OY f ∗OW ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' So for any generic point ξ ∈ Spec(A) corresponding to an irreducible component of f(Z) ∩ W, we have (f∗TorOY i (OZ, f ∗OW ))ξ = (TorOX i (f∗OZ, OW ))ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) On the other hand, by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='9 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='29, we have f∗([Z] · f ∗[W]) = � i (−1)if∗[TorOY i (OZ, f ∗OW )]r+s−dimK Y = � i (−1)i[f∗TorOY i (OZ, f ∗OW )]r+s−dimK Y , f∗[Z] · [W] = [f∗OZ] · [W] = � i (−1)i[TorOX i (f∗OZ, OW )]r+s−dimK Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then f∗([Z] · f ∗[W]) = f∗[Z] · [W] by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 7 GAGA It is natural to expect that our definitions of cycles, flat pull-backs, proper push-forwards and intersection products, for algebraic variety will be coincide with the ones in the intersection theory of algebraic varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let X be an algebraic variety over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then we have an isomorphism Z∗(X) ≃ Z∗(Xan), [Y ] �→ [Y an].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For a cycle α ∈ Z∗(X), we will denote its image in Z∗(Xan) by αan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Moreover, the following properties hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) For any affinoid domain V contained in some affine open subset of Xan, the diagram diagram commutes: Z∗(X) � � Z∗(V) � Z∗(Xan) � Z∗(V ) , where V = Spec(OXan(V )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 23 (2) Let α, β ∈ Z∗(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then α = β ∈ Z∗(X) (or αan = βan ∈ Z∗(Xan)) if and only if i∗α = i∗β ∈ Z∗(V) for any any affinoid domain V contained in some affine open subset of Xan, where V = Spec(OXan(V )) and i : V → X is the canonical morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The map is obviously injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' It is suffices to show that every integral closed subspace Z of Xan is algebraic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If X is proper over K, by GAGA result, see [2, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='11], we know that Z is algebraic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In general case, by Nagata’s compactification theorem, there is a proper variety X over K such that X ⊂ X is an open immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We take the Zariski-closure Z of Z in X an, which is algebraic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' there is an integral subvariety T ⊂ X such that T an = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We claim that (T ∩ X)an = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By construction of analytification, we have (T ∩ X)an = T an ∩ Xan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We also have Z ∩ Xan = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then T an = Z implies that (T ∩ X)an = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) The diagram is directly from the definition of [Y an] and Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='8 (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (2) This is from the isomorphism Z∗(X) ≃ Z∗(Xan), the commutative diagram in (1) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) We have a surjection CH∗(X) ։ A∗(Xan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let f : Y → X be a morphism of algebraic varieties over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We have the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) Let F be a coherent sheaf on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then [F]an = [Fan].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (2) We have a canonical homomorphism Div(X) → Div(Xan), D �→ Dan such that for any D ∈ Div(X), we have [D]an = [Dan].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (3) If ϕ is flat and α ∈ Z∗(X), then (ϕ∗(α))an = (ϕan)∗(αan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (4) If ϕ is proper and β ∈ Z∗(Y ), then (ϕ∗(β))an = (ϕan)∗(βan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (5) Let α, β ∈ Z∗(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then α, β intersect properly if and only if αan, βan ∈ Z∗(Xan) intersect properly, and in this case, we have (α · β)an = αan · βan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) Let V = M(B) ⊂ Xan be an affinoid domain contained in some affine open subsets of Xan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then we have a canonical morphism ϕ : Spec(A) → X which is flat by [7, TH´EOR`EM 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' It is sufficient to show that [F]an|V = [Fan]|V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By the commutative diagram in (1), we have [F]an|V = [ϕ∗F];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' by Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='8 (1), we have [Fan]|V = [Fan|V ] = [ϕ∗F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' So our claim holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (2) The homomorphism is given by the fact that V → X is flat for any an affinoid domain V = M(A) ⊂ Xan contained in some affine open subsets of Xan, where V = Spec(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then the compatibleness on such affinoid domains will induce a divisor on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The equality can be proved as (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (3) We take any affinoid domains V = M(A) ⊂ Xan and W = M(B) ⊂ Y an such that ϕan(W) ⊂ V and V , W are contained in some affine open subsets of Xan, Y an respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let V = Spec(A), W = Spec(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We have the following commutative diagram W j � �ϕ � V i � Y ϕ � X Then (ϕ∗(α))an|W = j∗ϕ∗(α) = �ϕ∗i∗(α) = (ϕan|W )∗(αan|V ) = (ϕan)∗(αan)|W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' here we identify the canonical isomorphisms Z∗(V ) ≃ Z∗(V) and Z∗(W) ≃ Z∗(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='5, (3) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (4) Since ϕ is proper, we have ϕan is proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We may assume that β is prime, moreover, assume that X, Y are integral and β = [X], ϕ is finite, surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Hence we can assume that X = Spec(A) and Y = Spec(B) are affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let V = M(A′) ⊂ Xan be an affinoid domain, and 24 U = (ϕan)−1(V ) = M(A′ ⊗A B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Notice that Frac(B) = B ⊗A Frac(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We consider the following diagram Frac(A) ⊗A A′ � Frac(B) ⊗A A′ Frac(A) � � Frac(B) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Notice that Frac(A) → Frac(B) is finite, so Frac(A) ⊗A A′ → Frac(B) ⊗A A′ is finite and flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We have that [Frac(B) : Frac(A)] = � q,ϕ(q)=p [(Frac(B) ⊗A A′)q : (Frac(A) ⊗A A′)p] where q runs through the minimal ideal of Frac(B)⊗A A′, and we view ϕ : Spec(Frac(B)⊗A A′) → Spec(Frac(A) ⊗A A′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The right-handed side is exactly deg(Y an/Xan) defined in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='23, so (4) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (5) We can assume that α, β are prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Since flat pull-backs preserve proper intersection, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='3, we know that α, β intersect properly if and only if αan, βan ∈ Z∗(Xan) intersect properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The proof of the equality is similar with the proof of (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 8 The category of finite correspondences In this section, we will define the additive category CorK of finite correspondences of K-analytic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We will follow the notation in [1] and the idea in [13, Lecture 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For the K-analytic spaces in this section, we always mean separated, quasi-paracompact, strictly K-analytic spaces, the category of such spaces is exactly the category of separated, quasi- paracompact, K-rigid spaces by [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' A K-analytic space is said to be quasi-smooth if it is geometrically regular at each point, see [8, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' In particular, a quasi-smooth space is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let X be a quasi-smooth, connected K-analytic space, and Y any K-analytic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' An elementary correspondence from X to Y is an irreducible closed subset W of X ×Y whose associated integral subspace is finite and surjective over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By an elementary corresponding from a quasi-smooth non-connected K-analytic space X to Y , we mean an elementary correspondence from a connected component of X to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The group CorK(X, Y ) is the free abelian group generated by the elementary correspondences from X to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The element of CorK(X, Y ) are called finite correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) If X is quasi-smooth, K-analytic space, one important example of elementary correspondence from X to Y is the graph Γf of a morphism f : X → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' If X is not connected, the Γf is a finite correspondence from X to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Notice that Γf is closed in X × Y since Y is separated and Γf is a section of X × Y → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (2) If X is not connected and X = � Xi is the decomposition into its connected components, we have CorK(X, Y ) = � i CorK(Xi, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (3) Every closed subspace Z of X × Y which is finite and surjective over X determines a finite correspondence [Z] from X to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We only consider the case where X is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We can write [Z] = � i ni[Zi], where Zi are irreducible component of Z such that Zi → X is surjective, and ni is the geometric multiplicity of Zi of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' To define the composition of morphism in the category CorK, we need the following lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let f : T → T ′ be a morphism of K-analytic spaces over another K-analytic space S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let W be an irreducible Zariski-closed subset of T which is finite and surjective over S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then f(W) is irreducible, Zariski-closed in T ′ and finite, surjective over S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Since T ′ → S is separated, W → S is finite, hence proper by [2, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='8], we know that W → T ′ is proper, see [5, Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' So f(X) is irreducible Zariski-closed in T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We replace T, T ′ by W, f(W) respectively, so we assume that T is finite and surjective over S, and surjective on T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By [2, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='8], it remains to show that T ′ is proper over S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Obviously T ′ → S is quasi-compact since T → T ′ is surjective and T ′ → S quasi-compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By [2, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='8 (iii)], we have T = Int(T/S) = Int(T/T ′) ∩ f −1(Int(T ′/S)) = f −1(Int(T ′/S)), this implies that Int(T ′/S) = T ′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' ∂(T ′/S) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' So T ′ is proper over S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let Z be an integral K-analytic space, finite and surjective over a normal K-analytic space S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then for every morphism S′ → S with S′ connected (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' irreducible), every connected (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' irreducible) component of Z ×S S′ is finite and surjective over S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This is in fact an algebraic result from [15, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' We can assume that S = M(A), Z = M(B) and S′ = M(A′) are affinoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Since B is finite over A, so B′ := B �⊗AA′ = B ⊗A A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By [15, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='17 (3)], we know that Spec(B) → Spec(A) is universally equidimensional, hence universally open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then Spec(B′) → Spec(A′) is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For every connected component T = M(C) of M(B′), the morphism Spec(C) → Spec(B′) is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' So M(C) → M(B′) has image that is closed and Zariski-open, which is exactly M(B′) since it is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' For the irreducible case, since Spec(B′) → Spec(A′) is equidimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then the image of each irreducible component Spec(C) of Spec(B′) is Spec(A′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Since the image of M(C) is a Zariski- closed subspace of M(A), it must be M(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let X, Y, Z be K-analytic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let V ⊂ X × Y and W ⊂ Y × Z be integral closed subspace which are finite and surjective over X and Y respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Assume that Y is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then V × Z and X × W intersect properly in X × Y × Z, and each component of the push-forward of the cycle [V × Z] · [X × W] on X × Z is finite and surjective over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Notice that V ×Y W ֒→ X × Y ×Y Y × Z ≃ X × Y × Z is the intersection of V × Z and X × W in X × Y × Z, see the explanation in the remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Then we have the following diagram V ×Y W � � W � � Z V � � Y X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='4, each component of V ×Y W is finite and surjective over V , so it is also finite and surjective over X, and it is of dimension dim X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This implies that V × Z and X × W intersect properly in X × Y × Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='3, the image of each component of V ×Y W in X × Z is finite and surjective over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Let CorK be the category defined as follows: Objects: the quasi-smooth K-analytic spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Morphisms: the finite correspondences CorK(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Given V ∈ CorK(X, Y ), W ∈ CorK(Y, Z), we define W ◦V as the push-forward of [V ×Z]·[X ×W] on X × Z, which is an element in CorK(X, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1) The composition is associative and bilinear, and the diagonal ∆X is the iden- tity for a quasi-smooth K-analytic space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This is from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='28 and Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='2, see the proof of [9, Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='1] for the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 26 (2) It is not hard to show that the category QSmK of quasi-smooth K-analytic spaces is fully faithful subcategory of CorK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (3) By [1, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='35] and a few work, we can see our definition of CorK coincide with [1, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Following the idea in [4], we can define higher Chow groups CHn(X, s) for quasi-smooth K- analytic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' By GAGA principle, such definition will coincide with the one for algebraic varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' On the other hand, the higher Chow groups is also defined in [1, Introduction g´en´erale] using motives of analytic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' It is natural to expect there is a close connection between these two and higher Chow groups have similar properties as in the case of algebraic varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Acknowledgements The author would like to thank my host professor, Yigeng Zhao for his encouragement, support and valuable suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' He would also like to thank Antoine Ducros, Walter Gubler and Michael Temkin for their patience and answering questions during his study of Berkovich spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' This research is supported by postdoctoral research grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' References [1] Ayoub, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Motifs des vari´et´es analytiques rigides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' M´em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' ), (140- 141):vi+386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' [2] Berkovich, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Spectral theory and analytic geometry over non-Archimedean fields, volume 33 of Mathematical Surveys and Monographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' American Mathematical Society, Provi- dence, RI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' [3] Berkovich, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' ´Etale cohomology for non-Archimedean analytic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Hautes ´Etudes Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=', (78):5–161 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' [4] Bloch, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Algebraic cycles and higher K-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=', 61(3):267–304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' [5] Bosch, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=', G¨untzer, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=', and Remmert, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Non-Archimedean analysis, volume 261 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Springer-Verlag, Berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' A systematic approach to rigid analytic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' [6] Ducros, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Variation de la dimension relative en g´eom´etrie analytique p-adique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Compos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=', 143(6):1511–1532.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' [7] Ducros, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Les espaces de Berkovich sont excellents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Fourier (Grenoble), 59(4):1443–1552.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' [8] Ducros, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Families of Berkovich spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Ast´erisque, (400):vii+262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' [9] Fulton, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Intersection theory, volume 2 of Ergebnisse der Mathematik und ihrer Grenzgebiete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Folge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 3rd Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' A Series of Modern Surveys in Mathematics].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Springer-Verlag, Berlin, second edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' [10] Grothendieck, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' ´El´ements de g´eom´etrie alg´ebrique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' ´Etude locale des sch´emas et des morphismes de sch´emas IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Hautes ´Etudes Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=', (32):361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' [11] Gubler, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Local heights of subvarieties over non-Archimedean fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Reine Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=', 498:61–113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' [12] Liu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Algebraic geometry and arithmetic curves, volume 6 of Oxford Graduate Texts in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Oxford University Press, Oxford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Translated from the French by Reinie Ern´e, Oxford Science Publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' 27 [13] Mazza, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=', Voevodsky, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=', and Weibel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Lecture notes on motivic cohomology, volume 2 of Clay Mathematics Monographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' American Mathematical Society, Providence, RI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Clay Mathematics Institute, Cambridge, MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' [14] Stacks project authors, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' The stacks project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' https://stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='columbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' [15] Voevodsky, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=', Suslin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=', and Friedlander, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Cycles, transfers, and motivic homology theories, volume 143 of Annals of Mathematics Studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Princeton University Press, Princeton, NJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content=' Cai, Westlake University, Dunyu Road 600, Xihu District 310024, Hangzhou, China E-mail address: caiyulin@westlake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} +page_content='cn 28' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQfwQFU/content/2301.02629v1.pdf'} diff --git a/8tE2T4oBgHgl3EQflgdv/vector_store/index.pkl b/8tE2T4oBgHgl3EQflgdv/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..dae58d5901172ea47f2193998832dd157a3e15c1 --- /dev/null +++ b/8tE2T4oBgHgl3EQflgdv/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8595b81e215466951b934e1321672553ed602fa0a57e1e0ff552d2ee2e07ae77 +size 194851 diff --git a/99AzT4oBgHgl3EQf_P4t/content/tmp_files/2301.01944v1.pdf.txt b/99AzT4oBgHgl3EQf_P4t/content/tmp_files/2301.01944v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0fc8c7d0bd5e83b2f79b32f6a21fccd1453e51df --- /dev/null +++ b/99AzT4oBgHgl3EQf_P4t/content/tmp_files/2301.01944v1.pdf.txt @@ -0,0 +1,3993 @@ +Draft version January 6, 2023 +Typeset using LATEX twocolumn style in AASTeX63 +Study of variability in long-term multiwavelength optical lightcurves of blazar AO 0235+164 +Abhradeep Roy +,1 Alok C. Gupta +,2, 3 Varsha R. Chitnis +,1 Sergio A. Cellone +,4, 5 Claudia M. Raiteri +,6 +Gustavo E. Romero +,7, 5 Paul J. Wiita +,8 Anshu Chatterjee +,1 Jorge A. Combi +,5, 7, 9 Mai Liao +,10, 11 +Arkadipta Sarkar +,12 and Massimo Villata +6 +1Department of High Energy Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-400005, India +2Aryabhatta Research Institute of Observational Sciences (ARIES), Manora Peak, Nainital 263001, India +3Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai +200030, China +4Complejo Astron´omico El Leoncito (CASLEO, CONICET-UNLP-UNC-UNSJ), San Juan, Argentina +5Facultad de Ciencias Astron´omicas y Geof´ısicas, Universidad Nacional de La Plata, La Plata, Buenos Aires, Argentina +6INAF-Osservatorio Astrofisico di Torino, Via Osservatorio 20, I-10025 Pino Torinese, Italy +7Instituto Argentino de Radioastronom´ıa (CCT-La Plata, CONICET; CICPBA; UNLP), Buenos Aires, Argentina +8Department of Physics, The College of New Jersey, 2000 Pennington Rd., Ewing, NJ 08628-0718, USA +9Deptamento de Ingenier´ıa Mec´anica y Minera, Universidad de Ja´en, Campus Las Lagunillas s/n Ed. A3 Ja´en, 23071, Spain +10CAS Key Laboratory for Researches in Galaxies and Cosmology, Department of Astronomy, University of Science and Technology of +China, Hefei, Anhui 230026, China +11School of Astronomy and Space Science, University of Science and Technology of China, Hefei, Anhui 230026, China +12Deutsches Elektronen-Synchrotron, Platanenallee 6, D-15738 Zeuthen, Germany +Submitted to ApJS +ABSTRACT +We present a long-term and intraday variability study on optical multiwaveband (UBVRI) data +from the blazar AO 0235+164 collected by various telescopes for ∼44 years (1975–2019). The blazar +was found to be significantly variable over the years in all wavebands with a variation of about six +magnitudes between its low and active states. The variations in the different wavebands are highly +correlated without any time-lag. We did not observe any significant trend in color variation with time, +but we observed a bluer-when-brighter trend between the B − I color index and the R-magnitude. +Optical BVR-band spectral energy distributions always show a convex shape. Significant intraday +variability was frequently seen in the quasi-simultaneous observations of AO 0235+164 made on 22 +nights in R and V -bands by the CASLEO and CAHA telescopes during 1999–2019. We also estimated +the central supermassive black-hole mass of 7.9 × 107M⊙ by analyzing the broad Mg II emission line +in AO 0235+164’s spectrum. We briefly explore the probable physical scenarios responsible for the +observed variability. +Keywords: galaxies: active – BL Lacertae objects: general – quasars: individual – BL Lacertae objects: +individual: AO 0235+164 +1. INTRODUCTION +Blazars belong to the radio-loud (RL) class of active +galactic nuclei (AGNs). This extremely variable class +is the union of BL Lacertae objects (BL Lacs) and +flat spectrum radio quasars (FSRQs). +Blazars host a +Corresponding author: Abhradeep Roy +abhradeep.1996@gmail.com, abhradeep.roy@tifr.res.in +large-scale relativistic jet of plasma pointing very close +to the observer’s line of sight (Urry & Padovani 1995). +The jet is launched from the very near vicinity of the +supermassive black hole (SMBH) of mass 106 – 1010 +M⊙ at the center of the AGN (e.g., Woo & Urry 2002). +Blazars are characterized by highly variable emission +throughout the whole electromagnetic (EM) spectrum, +from radio to γ-rays, and their spectral energy distri- +butions (SEDs) are characterized by two broad humps +(Fossati et al. 1998). +Blazars display high and vari- +arXiv:2301.01944v1 [astro-ph.HE] 5 Jan 2023 + +ID2 +Roy et al. +time (JD) +12 +13 +14 +15 +16 +17 +18 +19 +I mag +WEBT-GASP +CASLEO-CAHA +12 +14 +16 +18 +20 +R mag +WEBT-GASP +Hagen-Thorn et al. 2008 +SMARTS +Steward +Takalo et al. 1998 +CASLEO-CAHA +14 +15 +16 +17 +18 +19 +20 +V mag +WEBT-GASP +CASLEO-CAHA +SMARTS +Steward +14 +15 +16 +17 +18 +19 +20 +21 +B mag +WEBT-GASP +CASLEO-CAHA +SMARTS +2444000 +2446000 +2448000 +2450000 +2452000 +2454000 +2456000 +2458000 +Time (JD) +16 +17 +18 +19 +20 +21 +U mag +WEBT-GASP +1980 +1990 +2000 +2010 +2020 +Time (Year) +Figure 1. Long-term multiwavelength optical (U, B, V , R, I) lightcurves of AO 0235+164 observed from multiple ground-based +telescopes between JD 2442689 (1975 October 3) and JD 2458835 (2019 December 17). + +AO 0235+164 optical variability +3 +able polarization from radio to optical bands, and emit +predominately non-thermal emission in the entire EM +spectrum. +The low-energy hump is ascribed to syn- +chrotron radiation from relativistic leptons, whereas the +high-energy hump arises from inverse Compton (IC) +processes and sometimes from hadronic processes (e.g., +Marscher 1983; M¨ucke et al. 2003; Romero et al. 2017, +and references therein). +Blazars display flux variability on diverse timescales +ranging from a few minutes to several years. +Blazar +variability has often been divided into three categories, +depending on the cadence of the observations: (i) mi- +crovariability (Miller et al. 1989), or intraday variability +(IDV) (Wagner & Witzel 1995), or intra-night variabil- +ity (INV) (Sagar et al. 2004), focusing on the variability +over a day or less; (ii) short-term variability (STV), +focusing on variability over days to weeks, (iii) and +long-term variability (LTV), focusing on timescales of +months to years (e.g. Gupta et al. 2004). +The BL Lac object AO 0235+164 is at redshift z = +0.94 (Cohen et al. 1987). +Optical spectroscopic and +photometric observations of the object have discovered +two foreground-absorbing systems at z = 0.524 and z = +0.851 (Cohen et al. 1987; Nilsson et al. 1996; Raiteri +et al. 2007). +The flux of the source can be both ab- +sorbed and contaminated by these foreground systems, +and the stars in them may act as gravitational micro- +lenses that could contribute to the observed variability. +Abraham et al. (1993) did deep CFHT imaging of AO +0235+164 and reported that the source is weakly am- +plified by macrolensing / microlensing by stars in the +foreground. +AO 0235+164 has been extensively observed in the past +from radio to γ-ray bands either in individual EM bands +or quasi-simultaneously in multiple EM bands and has +shown variations in all those bands on diverse timescales +(e.g., Madejski et al. 1996; Rabbette et al. 1996; Takalo +et al. 1998; Qian et al. 2000; Webb et al. 2000; Romero +et al. 2000; Raiteri et al. 2006, 2008; Hagen-Thorn et al. +2008; Gupta et al. 2008; Agudo et al. 2011; Ackermann +et al. 2012; Fan et al. 2017; Kutkin et al. 2018; Wang +& Jiang 2020, and references therein). It is one of the +blazars which has displayed very high and variable op- +tical/NIR polarization up to ∼45 percent (e.g., Impey +et al. 1982; Stickel et al. 1993; Fan & Lin 1999; Cellone +et al. 2007; Ikejiri et al. 2011; Itoh et al. 2016, and +references therein). In the Hamburg quasar monitoring +program (HQM) this source was observed in the optical +R band during 1988–1993, during which a 2.36±0.25 +magnitude variation was detected; a particularly strong +brightening in the source of ∼1.6 magnitude was re- +ported during February 20–22, 1989 (Schramm et al. +1994). In six nights of optical B and V bands obser- +vations during 21–27 September 1992, the blazar was +found in an unusually bright state and IDV was de- +tected in both B and V bands (Rabbette et al. 1996). +On another occasion, 6 nights of quasi-simultaneous V +and R band observations in November 1999, revealed +IDV with an amplitude of ∼100 percent over timescales +of a day, while 0.5 magnitude changes were reported +in both bands on a single night (Romero et al. 2000). +In multicolor optical/NIR photometric (BVRIJHK) +and R-band optical polarimetric observations of AO +0235+164 during its 2006 December outburst, variabil- +ity on IDV timescales was detected, with increasing +minimum timescale of variability from optical to NIR +wavelengths; such variations were even detected in the +optical polarization (Hagen-Thorn et al. 2008). In three +nights of optical observations of the blazar in January – +March 2007, IDV and STV were detected (Gupta et al. +2008). +In quasi-simultaneous optical (V and R bands) and +radio (22 GHz) observations of AO 0235+164 during +1993–1996, the variability in optical bands showed am- +plitudes up to 1.5 magnitudes on STV timescales; al- +though the radio variability is less dramatic, in general, +it followed the optical behavior (Takalo et al. 1998). For +the 1997 AO 0235+164 outburst, quasi-simultaneous +multi-wavelength (MW) (radio, optical, NIR, and X- +ray) observations were carried out. It was found that +the source varied nearly simultaneously over 6 decades +in frequency during the outburst and this result was +explained in terms of a microlensing event (Webb et al. +2000). +An analysis of this source’s variability over ∼25 years +led to the suggestion of a ∼5.7 years quasi-periodicity +of the main radio and optical flares (Raiteri et al. 2001); +however, the putative next outburst, predicted to peak +around February–March 2004, did not occur, and a +new analysis of the optical light curves on a longer +time span revealed a characteristic variability timescale +of ∼8 years, which was also present in the radio data +(Raiteri et al. 2006). Recently, optical R band photo- +metric data taken during 1982–2019 showed 5 cycles +of double-peaked periodicity of ∼8.13 years with a sec- +ondary peak following the primary one by ∼(1.5–2.0) +years (Roy et al. 2022). In another MW campaign from +radio to UV bands in 2006–2007, a huge NIR-optical- +UV outburst with brightness increase of ∼5 magnitudes + +4 +Roy et al. +during February 19 – 21, 2007 was detected (Raiteri +et al. 2008). +During a major outburst seen in 2009, +changes in radio, optical, X-ray, and γ-ray bands were +found to be strongly associated (Agudo et al. 2011). +In another simultaneous MW observing campaign of +this blazar between 2008 September and 2009 February, +γ-ray activity was found to be well correlated with a se- +ries of NIR/optical flares, accompanied by an increase in +the optical degree of polarization; the X-ray light curve +showed a different 20-day high state of an unusually +soft spectrum which did not match the extrapolation +of the optical/UV synchrotron spectrum (Ackermann +et al. 2012). +AO 0235+164 is one of the sources that often used +to be called OVV (optically violently variable). There +are several such objects, like 3C 279, 3C 454.3, 4C +29.45, CTA 102, BL Lacertae, etc. +Long-term achro- +maticity and zero lags have widely been found for these +sources (Bonning et al. 2012; Zhang et al. 2021; Fan +et al. 2006; Raiteri et al. 2017; Guo et al. 2015). AO +0235+164 is peculiar because it is commonly considered +a BL Lac, one of the furthest known, but it shares +properties with FSRQs. +It is also a complex source +because its light is contaminated by the southern AGN, +ELISA, and absorbed by an intervening galaxy. This +paper has undertaken a detailed analysis of the source’s +optical brightness and spectral variability over a very +long time span (∼5 decades) as well as an investiga- +tion of its central engine. Our aim is to shed light on +the long and short-term behavior of an emblematic BL +Lac object through a detailed analysis of what is likely +the most massive data set ever assembled for an object +of this kind. +The paper is organized as follows. +In +section 2, we provide descriptions of the observations +of AO 0235+164. The section 3 gives our data analy- +sis methods and results. We present a discussion and +conclusions in section 4 and section 5, respectively. +2. OBSERVATIONS +Most of the optical UBV RI +observations of AO +0235+164 we have employed in this work are taken +from The Whole Earth Blazar Telescope1 (WEBT) +(Villata et al. 2002; Raiteri et al. 2017) which is an in- +ternational collaboration of optical, near-infrared, and +radio observers. WEBT has organized several monitor- +ing campaigns on the blazar AO 0235+164, with the +participation of many tens of observers and telescopes +all around the world. +Later, this source was studied +1 https://www.oato.inaf.it/blazars/webt +by the WEBT and by its GLAST-AGILE Support Pro- +gram (GASP) (Villata et al. 2008, 2009), which was +started in 2007 to record quasi-simultaneous data of +various blazars observed by the AGILE and Fermi (for- +merly GLAST) satellites. WEBT/GASP data on AO +0235+164 were published in Raiteri et al. (2001, 2005, +2006, 2008) and Ackermann et al. (2012). Raiteri et al. +(2005) prescribed ways to remove the contribution of +the southern galaxy ELISA from the observed optical +flux densities and estimated the amount of absorption +towards the source in excess of that from our Galaxy in +X-ray, ultraviolet, optical, and near-infrared bands. +The WEBT and GASP data were calibrated following +a common prescription, i.e., with the same photome- +try for the same reference stars. For calibration of the +AO 0235+164 observations, the adopted photometric +sequence includes stars 1, 2, and 3 from Smith et al. +(1985). To build a reliable lightcurve for further anal- +ysis, clear outliers were removed and minor systematic +offsets between various datasets were corrected. +AO 0235+164 was also observed with the 2.2 m tele- +scope of Calar Alto Astronomical Observatory (CAHA, +Spain) in November – December 2005, using the CAFOS +instrument in imaging polarimetry mode, and photo- +metric data were obtained by adding up the ordinary +and extraordinary fluxes from each individual image +(Cellone et al. 2007). +Photometric data were also +obtained with the 2.15 m telescope at Complejo As- +tron´omico El Leoncito (CASLEO, Argentina) along +several runs in November 1999, December 2000, August +2004, and January 2005. Results from these data were +published in Romero et al. (1999, 2000, 2002) and in +two papers by the WEBT collaboration focused on this +blazar (Raiteri et al. 2005, 2006). +Data from a more +recent (December 2019) observing run with the same +telescope were used in Roy et al. (2022). +Magnitude +calibration to the standard system was done using our +own photometry of Landolt’s (2009) fields as well as +standard stars in the field of AO 0235+164 (Smith +et al. 1985; Gonz´alez-P´erez et al. 2001). +We also collected the publicly available optical R and +V -band data of AO 0235+164, taken at Steward Ob- +servatory2, University of Arizona. These measurements +employed the 2.3 m Bok and 1.54 m Kuiper telescopes +between 4 October 2008 and 12 February 2018, using +the SPOL CCD Imaging/Spectropolarimeter attached +2 http://james.as.arizona.edu/∼psmith/Fermi/DATA/Rphotdata. +html + +AO 0235+164 optical variability +5 +Table 1. Result of flux variability on optical UBVRI long-term +lightcurves of AO 0235+164 +Optical +Total +χ2 +red. +χ2 +0.999,red. +Status +Variability +filter +Obs. +amplitude (%) +U +109 +904.5 +1.47 +V +548.8 +B +894 +3246.7 +1.15 +V +590.9 +V +1403 +5968.4 +1.12 +V +589.0 +R +5675 +8715.5 +1.06 +V +718.8 +I +1173 +3555.2 +1.13 +V +567.5 +Note—In the fourth column ’V/NV’ represents variable/non- +variable status. +to those two telescopes. +Details about the instru- +ment, observation, and data analysis are given in Smith +et al. (2009). +In addition, we included the optical- +BV R data from the Small and Moderate Aperture +Research Telescope System (SMARTS) public archive3. +The SMARTS consortium is part of the Cerro Tololo +Inter-American Observatory (CTIO), Chile, and has +been observing Fermi-Large Area Telescope (LAT)- +monitored blazars in the optical B, V , R and NIR J +and K bands. Details about the SMARTS instruments, +observations, and data analysis procedures are given +in Bonning et al. (2012). These standard magnitudes +observed by CASLEO, CAHA, SMARTS, and the Stew- +ard observatory were further corrected for the southern +galaxy ELISA following Raiteri et al. (2005). We also +added other R-band optical photometric data from the +literature (Takalo et al. 1998; Hagen-Thorn et al. 2008). +3. DATA ANALYSIS METHODS AND RESULTS +We combined all the optical U, B, V , R, I band data +to plot the long term (1974–2020) MW lightcurves of +blazar AO 0235+164 (Figure 1). We removed the ob- +servations with errors of more than 0.1 magnitudes and +studied long-term and intraday variability, color varia- +tion, spectral properties, and inter-band correlations. +3.1. Flux variability studies +We use different tools on the observed optical magni- +tudes to quantify the variability timescales and the cor- +responding significance in multiple optical wavebands. +3.1.1. The χ2test +3 http://www.astro.yale.edu/smarts/glast/home.php# +For a time series of flux density observations, the χ2 is +defined as, +χ2 = +N +� +i=1 +(Mi − ¯ +M)2 +ε2 +i +(1) +where Mi is the magnitude obtained at the ith observa- +tion, εi is the corresponding error in measurement, and +¯ +M is the average magnitude. If the obtained χ2 value +is higher than the critical χ2 value at 99.9 per cent sig- +nificance level, we consider the source as variable. The +critical value (χ2 +0.999,d) depends on the degrees of free- +dom (d) of the dataset. The reduced χ2 values listed in +Table 1 indicate that the source exhibits significant flux +variations in all the optical wavebands. +3.1.2. Variability amplitude +According to the relation given by Heidt & Wagner +(1996), we estimated the variability amplitudes (VM) in +percentage for the lightcurves in different wavelengths +using the following formula, +VM = 100 × +� +(Mmax − Mmin)2 − 2 ¯ε2 (%) +(2) +where Mmax and Mmin are the maximum and minimum +observed magnitude in a lightcurve, respectively, while +¯ε is the average error in magnitude measurements. We +list the calculated variability of amplitudes in Table 1. +3.1.3. Correlation study +To study the inter-band correlations, we first gener- +ated 15-minute binned optical UBVRI lightcurves, and +plotted the average U, B, V , and I-magnitudes against +the average R-magnitudes for the time bins when the +source was observed at both the wavebands (Figure 2). +The magnitude-vs-magnitude plots show very good +linear correlations. To take the uncertainty of magni- +tude measurements into account, we simulated 10000 +datasets assuming that each magnitude measurement +is Gaussian distributed. Then we calculated the mean +and standard deviation of the Pearson correlation co- +efficients of all simulated datasets. We obtained high +correlations (> 0.9) with small uncertainties (< 0.003) +between all wavebands. +Moreover, to find any time lag between the correlated +optical lightcurves we computed the discrete correlation +function (DCF) from the unbinned multiwavelength +light curves, as the light curves consist of discrete data +points. +Following the method of Edelson & Krolik +(1988), we computed the unbinned DCF (UDCF) be- + +6 +Roy et al. +15 +16 +17 +18 +R magnitude +17 +18 +19 +20 +U magnitude +U-mag vs R-mag +Pearson coeff. = 0.96±2.93e-03 +fit: Umag = 0.92*Rmag+3.24 +14 +15 +16 +17 +18 +19 +R magnitude +15 +16 +17 +18 +19 +20 +V magnitude +V-mag vs R-mag +Pearson coeff. = 0.99±2.65e-04 +fit: Vmag = 1.00*Rmag+0.79 +14 +15 +16 +17 +18 +19 +R magnitude +16 +17 +18 +19 +20 +21 +B magnitude +B-mag vs R-mag +Pearson coeff. = 0.99±4.30e-04 +fit: Bmag = 1.01*Rmag+1.65 +14 +15 +16 +17 +18 +19 +R magnitude +13 +14 +15 +16 +17 +18 +19 +I magnitude +I-mag vs R-mag +Pearson coeff. = 0.99±2.58e-04 +fit: Imag = 0.98*Rmag-0.64 +Figure 2. +15-minute averaged UBV I magnitudes versus R-magnitude plots for correlation study. +U, B, V , and I-band +observations show high linear correlation with R-band data. All the plots are fitted with straight lines. +tween the ith data point in one waveband (a) and the +jth data point in another (b) as +UDCFij = (ai − ¯a)(bj − ¯b) +σaσb +, +(3) +where ¯a and ¯b are the mean of the observed magnitudes, +and σa and σb are the standard deviations of the cor- +responding datasets. Next, we calculated the discrete +correlation function (DCF) at a certain time lag τ by +averaging the UDCFijs whose corresponding time lags +∆tij = ta +i − tb +j lie within the range [τ − ∆τ +2 , τ + ∆τ +2 ] (∆τ +is the time lag bin width), such that, +DCF(τ) = 1 +n +� +UDCFij(τ). +(4) +Following the suggestion of White & Peterson (1994), +we computed the mean magnitudes (¯a and ¯b) and the + +AO 0235+164 optical variability +7 +10.0 +7.5 +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +Time lag (days) +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +1.1 +1.2 +1.3 +DCF +U vs R +10.0 +7.5 +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +Time lag (days) +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +DCF +B vs R +10.0 +7.5 +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +Time lag (days) +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +DCF +V vs R +10.0 +7.5 +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +Time lag (days) +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +DCF +I vs R +Figure 3. Results of discrete cross-correlation analysis of U, B, V , and I-band with respect to R-band in the full time range. +standard deviations (σa and σb) in Equation 3 using only +those data points who fall within a given time lag bin, as +the mean and standard deviation keep on changing for a +time series originated from a stochastic process such as +blazar emission. The error in the DCF(τ) computation +in each bin is calculated as +σDCF(τ) = +1 +M − 1 +� +� +� +� +M +� +k=1 +(UDCFk − DCF(τ))2. +(5) +Figure 3 shows the DCFs of UBV I bands with respect +to the R-band observations. In all cases, the DCFs peak +at zero time lag, except the U-band vs R-band DCF +due to poor data sampling in the U-band. This explains +the strong linearity in Figure 2 and implies that the +emission at all optical wavebands are coming from the +same region in the jet and are produced from the same +radiation mechanism. +Table 2. Color variation with time in optical UBVRI long- +term lightcurves of AO 0235+164 +CI +m +c +ρ +p +U − B +−1.52E-05 +3.74E+01 +−2.06E-01 +8.28E-02 +B − V +6.58E-06 +−1.52E+01 +1.42E-01 +4.79E-03 +V − R +−5.34E-06 +1.38E+01 +−9.19E-02 +1.39E-02 +R − I +1.83E-05 +−4.40E+01 +2.85E-01 +1.74E-08 +U − I +5.63E-05 +−1.35E+02 +4.03E-01 +1.88E-03 +B − I +4.16E-05 +−9.92E+01 +4.50E-01 +3.41E-11 +Note—In the column headings: CI: color indices; m = slope; +c = intercept; ρ = Pearson coefficient; p = null hypothesis +probability for Figure 4a +3.1.4. Color Variations +The term ‘color’ denotes the magnitude difference be- +tween two quasi-simultaneous observations at two dif- + +8 +Roy et al. +1 +0 +1 +U-B +0 +1 +2 +B-V +0 +1 +2 +V-R +0.0 +1.5 +Color +R-I +1.5 +3.0 +4.5 +U-I +2444000 +2448000 +2452000 +2456000 +Time (JD) +1.5 +3.0 +B-I +(a) +1 +0 +1 +U-B +0.8 +1.6 +B-V +0 +1 +2 +V-R +0.0 +1.5 +Color +R-I +1.5 +3.0 +4.5 +U-I +14 +15 +16 +17 +18 +19 +20 +R magnitude +1.5 +3.0 +B-I +(b) +Figure 4. (a) Color variation with time. (b) Color variation with optical R magnitude. The red line in each panel represents +the straight line fit. Fit parameters are given in Table 2 and Table 3 respectively. +Table 3. Color variation with R-band magnitude in optical +UBVRI long-term lightcurves of AO 0235+164 +CI +m +c +ρ +p +U − B +−1.36E-01 +2.37E+00 +−5.37E-01 +3.35E-05 +B − V +1.62E-02 +7.04E-01 +1.41E-01 +7.41E-03 +V − R +−3.54E-03 +7.98E-01 +−2.58E-02 +4.92E-01 +R − I +1.62E-02 +7.00E-01 +1.37E-01 +7.59E-03 +U − I +−6.47E-02 +3.85E+00 +−2.07E-01 +1.30E-01 +B − I +6.23E-02 +1.66E+00 +3.66E-01 +1.69E-07 +Note—In the column headings: CI: color indices; m = +slope; c = intercept; ρ = Pearson coefficient; p = null +hypothesis probability for Figure 4b +ferent wavebands. We plotted the variation of optical +colors (U − B, B − V , V − R, R − I, and B − I) with +time and R-magnitude in Figure 4. We listed the re- +sults of a straight line (Y = mX + c) fitting to all these +plots in Table 2 and Table 3. The linear fits of the color +versus time plots do not show any trend, except for the +rather sparsely sampled (B − I) color, which has a high +slope (4.16×10−5) in Figure 4a, along with the highest +Pearson correlation coefficient (0.45), and the lowest null +hypothesis probability (3.41×10−11). Among the color +versus magnitude relations, the strongest relationship is +between (B − I) and R (Figure 4b), having a positive +slope (6.23×10−2) with the highest Pearson coefficient +(0.37) and the lowest p-value (1.69×10−7) (Table 3), in- +dicates a bluer-when-brighter (BWB) trend when the +widest range of the available colors is considered. +3.1.5. Spectral Variations and SEDs +We plotted the optical (BVR) spectral energy distri- +butions for the nights where observations were taken +at all of these three filters. Following the prescription +of Raiteri et al. (2005), we took into account the total +absorption by the Milky Way galaxy and the foreground + +AO 0235+164 optical variability +9 +Figure 5. An example frame of the AO 0235+164 optical SED animation that is available in the HTML version of this article. +The duration of the animation is 1 minute and it contains a total of 360 one-day averaged optical SEDs, having 6 SEDs per +frame. The observation dates of the SEDs are given in the plot legend. +Table 4. Spetral index variation with R-band magnitude and +time in optical UBVRI long-term lightcurves of AO 0235+164 +Dependency +m +c +ρ +p +αV R vs R +−2.01E-02 +3.30E+00 +−2.58E-02 +4.92E-01 +αV R vs JD +−3.03E-05 +7.74E+01 +−9.19E-02 +1.39E-02 +Note—In the column headings: m = slope; c = intercept; ρ = +Pearson coefficient; p = null hypothesis probability for Figure 7. +absorber at z = 0.524, and subtracted the extinction +magnitudes (AU = 2.519, AB = 1.904, AV = 1.473, +AR = 1.260, AI = 0.902) from the calibrated magni- +tudes of respective wavebands and then converted them +into extinction-corrected flux densities, Fν. The accom- +panying video contains one-day averaged optical SEDs +for those 360 nights (An example frame is shown in +Figure 5). Figure 6 shows a few examples of SEDs of +low, moderate, and high flux states, plotted in (νFν – +ν) format. +Mostly, the SEDs have a declining shape +following a power law. However, there are evidences of +spectral hardening on several nights (e.g., JD 2445337, +JD 2445721, JD 2448889, JD 2452901, JD 2453230). +From the one-day binned multiwavelength lightcurves +we calculated the spectral indices (αV R) for all the days +when the source was observed in both V and R bands, + +AO 0235+164 Optical SEDs +10-10 +10-11 +JD 2448265 +JD 2448266 +JD 2448268 +JD 2448269 +10-12 +JD 2448889 +JD 2449601 +5 × 1014 +6 × 1014 +7 × 1014 +V (HZ)10 +Roy et al. +5 × 1014 +6 × 1014 +7 × 1014 +8 × 1014 + (HZ) +10 +12 +10 +11 +10 +10 +F (erg cm +2 s +1) +JD 2449690 +JD 2452169 +JD 2445343 +JD 2451896 +JD 2457045 +JD 2450811 +JD 2454733 +JD 2446763 +JD 2453230 +Figure 6. Examples of AO 0235+164 optical intraday SEDs +during three different states of brightness: (i) the green lines +represent SED during quiescent states (νFν (erg cm−2 s−1) +< 10−12), (ii) the blue lines show SED during moderately +bright states (10−12 < νFν (erg cm−2 s−1) < 3×10−11), (iii) +the red lines show SED during outbursts (νFν (erg cm−2 +s−1) > 5×10−11). The black lines are examples of SED with +spectral hardening on JD 2446763 and JD 2453230. +using the formula given by Wierzcholska et al. (2015) +on extinction corrected magnitudes, as +αV R = 0.4(V − R) +log(νV /νR) , +(6) +where νV and νR respectively represent the effective fre- +quencies of V and R band filters (Bessell 2005). +We +plotted the variation of spectral indices with time and +R-band magnitude (Figure 7) and listed the results of +linear fits, Pearson coefficient, and null hypothesis prob- +ability in Table 4. We do not find any significant long- +term variation of the spectral index with time, nor is +there a correlation with R-magnitude. +3.2. Intraday Variability +We applied four frequently used statistical tests for IDV: +scaled C-criterion, scaled F-test, the power-enhanced F- +test, and the nested analysis of variance (ANOVA) test +(de Diego 2014; de Diego et al. 2015; Zibecchi et al. +2017, 2020) to detect statistically significant intraday +flux variability in AO 0235+164 lightcurves observed by +CASLEO and CAHA telescopes. +These tests mainly +compare the variations in blazar magnitudes with the +variations in magnitudes of one or more stars within +the field-of-view of the blazar and have different advan- +tages and disadvantages. We collected data from mul- +tiple field stars along with the blazar data (Table 5). +We applied the first three methods on the intraday dif- +ferential lightcurves of AO 0235+164 where at least 10 +observations were recorded per night with at least one +optical filter between 1999 November 2 to 2019 Decem- +ber 17. We employed the nested ANOVA test only on +lightcurves having at least 20 observations per night. +Table 5. Equivalence between internal field star numbering in +the CASLEO/CAHA data used in the IDV analyses and field- +star numbering in other standard star charts during different +observation seasons +Season +CASLEO/CAHA +Heidelberga +GKM2001b +1999–2001 +2 +8 +10 +(CASLEO) +4 +C1 +9 +5 +6 +11 +7 +– +1 +8 +– +3 +10 +– +8 +12 +– +16 +2004–2005 +2 +8 +10 +(CASLEO) +4 +C1 +9 +5 +6 +11 +6 +– +8 +7 +– +7 +2005 +2 +8 +10 +(CAHA) +11 +C1 +9 +12 +– +1 +13 +– +3 +14 +– +7 +15 +– +8 +16 +6 +11 +17 +– +16 +2018–2019 +2 +8 +10 +(CASLEO) +4 +C1 +9 +5 +6 +11 +6 +– +8 +7 +– +7 +8 +– +16 +Note—a. +https://www.lsw.uni-heidelberg.de/projects/ +extragalactic/charts/0235+164.html +b. Gonz´alez-P´erez et al. (2001) +3.2.1. Scaled C-criterion +Differential photometry, where the blazar magnitudes +are compared to one or more stars in the same field +of view, is the usual technique for obtaining blazar +lightcurves free from the effects of any non-astrophysical +fluctuations. +The simplest differential photometry in- +volves a single comparison star, while a second star, +whose magnitudes are measured against the same com- +parison star, is used for a stability check. We denote B, +S1, and S2 as the blazar, comparison, and control star, +respectively. The variability test requires two differen- +tial lightcurves (DLC): (blazar–comparison star) and +(control star–comparison star). The latter is believed + +AO 0235+164 optical variability +11 +Table 6. Result of scaled C-criterion and F-test for IDV on AO 0235+164 differential lightcurves from +CASLEO and CAHA +Date +JD +Band +No. of +S1, S2 +Γ +CΓ +FΓ +F 0.005 +c +Status +Final +obs. +Status +1999 Nov 2 +2451485 +V +23 +2,3 +0.8886 +11.3640 +129.1405 +3.1246 +V +V +2,6 +1.0867 +12.9184 +166.8856 +3.1912 +V +2,10 +1.6876 +8.1627 +66.6298 +3.1246 +V +2,11 +0.7431 +13.4002 +179.5650 +3.1246 +V +1999 Nov 3 +2451486 +V +22 +2,3 +1.0707 +5.6976 +32.4624 +3.1347 +V +V +2,11 +0.8841 +6.0726 +36.8768 +3.1347 +V +1999 Nov 4 +2451487 +R +30 +2,3 +1.0059 +8.4058 +70.6582 +2.6737 +V +V +2,11 +0.6639 +9.8857 +97.7278 +2.6737 +V +V +30 +2,3 +0.9994 +8.9281 +79.7104 +2.6737 +V +V +2,11 +0.8286 +9.6683 +93.4770 +2.6737 +V +1999 Nov 5 +2451488 +R +23 +2,3 +1.4994 +1.5631 +2.4433 +3.1246 +NV +NV +2,11 +0.9852 +1.9303 +3.7260 +3.1246 +NV +V +22 +2,3 +1.4403 +3.0342 +9.2064 +3.1347 +V +V +1999 Nov 6 +2451489 +R +30 +2,3 +0.8471 +17.5775 +308.9682 +2.6737 +V +V +2,6 +0.9769 +12.3281 +151.9824 +2.6737 +V +2,7 +1.3573 +9.9373 +98.7501 +2.7048 +V +2,8 +1.3805 +9.8381 +96.7876 +2.7048 +V +2,10 +1.6936 +6.8657 +47.1376 +2.6737 +V +2,11 +0.5616 +15.4338 +238.2019 +2.6737 +V +V +29 +2,3 +0.8485 +18.1892 +330.8486 +2.7233 +V +V +2,6 +1.0013 +11.7527 +138.1254 +2.7233 +V +2,7 +1.3527 +12.5480 +157.4516 +2.7397 +V +2,8 +1.4133 +13.4172 +180.0214 +2.7397 +V +2,10 +1.5626 +17.6674 +312.1376 +2.7233 +V +2,11 +0.7018 +17.9948 +323.8145 +2.7233 +V +1999 Nov 7 +2451490 +R +11 +2,3 +0.9562 +3.5930 +12.9095 +5.8479 +V +PV +2,4 +1.9798 +2.2801 +5.1990 +5.8479 +NV +2,6 +1.1143 +4.3903 +19.2751 +5.8479 +V +2,10 +1.9703 +1.7073 +2.9148 +5.8479 +NV +2,11 +0.6197 +2.9496 +8.7003 +5.8479 +V +V +12 +2,3 +0.9382 +2.9304 +8.5871 +5.3191 +V +PV +2,4 +1.7807 +1.9342 +3.7410 +5.3191 +NV +2,6 +1.1169 +2.8931 +8.3701 +5.3191 +V +2,10 +1.7653 +2.1046 +4.4292 +5.3191 +NV +2,11 +0.7772 +4.3359 +18.7997 +5.3191 +V +Note—S1 and S2 are the comparison and control star numbers, respectively, used for the IDV tests. Star +numbers follow the star maps shown in Table 5. + +12 +Roy et al. +13 +14 +15 +16 +17 +18 +R magintude +2 +0 +2 +4 +6 +8 +10 +VR +-0.02*R+3.30 +(a) +2445000 +2447500 +2450000 +2452500 +2455000 +2457500 +Time (JD) +2 +0 +2 +4 +6 +8 +10 +VR +-3.03e-05*Time+7.74e+01 +(b) +Figure 7. (a) Variation of spectral index (αV R) with R-band magnitude. (b) Variation of αV R with time. The red line at each +panel represents the linear fit. +Table 6. Result of scaled C-test and F-test for IDV on AO 0235+164 differential lightcurves from CASLEO +and CAHA (continued...) +Date +JD +Band +No. of +S1, S2 +Γ +CΓ +FΓ +F 0.005 +c +Status +Final +obs. +status +2000 Dec 21 +2451900 +R +10 +2,3 +0.9446 +2.3638 +5.5876 +6.5402 +NV +PV +2,6 +1.0793 +4.9877 +24.8767 +6.5402 +V +2,7 +1.5020 +2.0187 +4.0753 +6.5402 +NV +2,8 +1.5289 +1.9985 +3.9939 +6.5402 +NV +2,9 +0.8790 +2.5120 +6.3100 +6.5402 +NV +2,11 +0.6246 +7.4168 +55.0085 +6.5402 +V +V +10 +2,3 +0.9509 +3.4671 +12.0208 +6.5402 +V +PV +2,6 +1.1202 +2.4789 +6.1449 +6.5402 +NV +2,7 +1.5357 +1.8729 +3.5079 +6.5402 +NV +2,8 +1.6064 +2.0031 +4.0124 +6.5402 +NV +2,9 +1.0966 +3.7299 +13.9120 +6.5402 +V +2,11 +0.7842 +1.5920 +2.5343 +6.5402 +NV +2000 Dec 23 +2451902 +R +10 +2,3 +0.8588 +4.4475 +19.7803 +6.5402 +V +V +2,6 +0.9890 +5.1629 +26.6559 +6.5402 +V +2,7 +1.3855 +3.5919 +12.9020 +6.5402 +V +2,8 +1.4091 +2.8222 +7.9646 +6.5402 +V +2,9 +0.8000 +4.6739 +21.8451 +6.5402 +V +2,11 +0.5664 +5.3690 +28.8267 +6.5402 +V +2,13 +1.7083 +3.0181 +9.1089 +6.5402 +V +V +11 +2,3 +0.8509 +6.5241 +42.5634 +5.8479 +V +PV +2,6 +1.0031 +5.4277 +29.4602 +5.8479 +V +2,7 +1.3714 +5.0139 +25.1395 +5.8479 +V +2,8 +1.4341 +5.2879 +27.9619 +5.8479 +V +2,9 +0.9797 +1.4805 +2.1919 +5.8479 +NV +2,11 +0.7013 +5.1765 +26.7965 +5.8479 +V +2,13 +1.5668 +4.3770 +19.1586 +5.8479 +V +Note—S1 and S2 are the comparison and control star numbers respectively used for the IDV tests. Star +numbers follow the star maps shown in Table 5. + +AO 0235+164 optical variability +13 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +JD (+2451485) +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +Differential magnitude +Date: 1999 Nov 02 +[Status: Variable] +V band +Blazar-S1 +(S2-S1) +0.45 +0.50 +0.55 +0.60 +0.65 +0.70 +JD (+2453680) +0.625 +0.650 +0.675 +0.700 +0.725 +0.750 +0.775 +0.800 +0.825 +Differential magnitude +Date: 2005 Nov 05 +[Status: Variable] +R band +Blazar-S1 +(S2-S1)+0.02 +0.54 +0.56 +0.58 +0.60 +0.62 +0.64 +0.66 +0.68 +JD (+2451900) +0.50 +0.52 +0.54 +0.56 +0.58 +0.60 +0.62 +0.64 +Differential magnitude +Date: 2000 Dec 21 +[Status: Probably Variable] +V band +Blazar-S1 +(S2-S1) +0.46 +0.48 +0.50 +0.52 +0.54 +0.56 +0.58 +JD (+2453711) +1.40 +1.42 +1.44 +1.46 +1.48 +Differential magnitude +Date: 2005 Dec 06 +[Status: Probably Variable] +R band +Blazar-S1 +(S2-S1)+0.69 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +JD (+2452225) +1.650 +1.675 +1.700 +1.725 +1.750 +1.775 +1.800 +Differential magnitude +Date: 2001 Nov 11 +[Status: Non Variable] +V band +Blazar-S1 +(S2-S1)+1 +0.56 +0.58 +0.60 +0.62 +0.64 +0.66 +JD (+2458835) +2.20 +2.22 +2.24 +2.26 +2.28 +2.30 +2.32 +2.34 +2.36 +Differential magnitude +Date: 2019 Dec 17 +[Status: Non Variable] +R band +Blazar-S1 +(S2-S1)+1.55 +Figure 8. Some intraday lightcurves of AO 0235+164 on nights when the source showed different states of variability. S1 and +S2 represent the comparison and control star respectively. In some panels, the differential lightcurve of the control star is shifted +to bring it into the same frame of the blazar DLC for better visual comparison of variability. + +14 +Roy et al. +Table 6. Result of scaled C-test and F-test for IDV on AO 0235+164 differential lightcurves from CASLEO +and CAHA (continued...) +Date +JD +Band +No. of +S1, S2 +Γ +CΓ +FΓ +F 0.005 +c +Status +Final +obs. +status +2001 Nov 9 +2452223 +R +12 +2,11 +1.2042 +4.6476 +21.5998 +5.3191 +V +V +V +12 +2,3 +1.8778 +2.5035 +6.2675 +5.3191 +NV +NV +2,4 +3.6191 +1.1450 +1.3111 +5.3191 +NV +2,9 +2.2039 +1.2380 +1.5326 +5.4171 +NV +2,10 +3.5871 +2.0056 +4.0226 +5.3191 +NV +2,11 +1.5366 +1.7566 +3.0857 +5.3191 +NV +2001 Nov 10 +2452224 +R +10 +2,3 +2.3728 +1.0570 +1.1172 +6.5402 +NV +NV +2,9 +2.2429 +1.1058 +1.2229 +6.5402 +NV +2,11 +1.5595 +0.9395 +0.8826 +6.5402 +NV +V +10 +2,3 +2.3876 +1.0788 +1.1637 +6.5402 +NV +NV +2,9 +2.7847 +1.3860 +1.9209 +6.5402 +NV +2,11 +1.9713 +0.9038 +0.8168 +6.5402 +NV +2001 Nov 11 +2452225 +R +14 +2,3 +2.0291 +1.4125 +1.9951 +4.5724 +NV +NV +2,9 +1.5447 +1.2505 +1.5638 +4.6425 +NV +2,11 +1.3171 +1.6860 +2.8427 +4.5724 +NV +V +14 +2,3 +2.0291 +1.4125 +1.9951 +4.5724 +NV +NV +2,9 +1.5447 +1.2505 +1.5638 +4.6425 +NV +2,11 +1.3171 +1.6860 +2.8427 +4.5724 +NV +2001 Nov 12 +2452226 +R +12 +2,3 +1.8479 +1.5819 +2.5025 +5.3191 +NV +PV +2,11 +1.2074 +3.0203 +9.1222 +5.3191 +V +V +12 +2,3 +1.8704 +1.9230 +3.6980 +5.3191 +NV +NV +2,4 +3.5981 +1.0281 +1.0571 +5.3191 +NV +2,10 +3.5672 +2.3374 +5.4634 +5.3191 +NV +2,11 +1.5330 +1.5642 +2.4468 +5.3191 +NV +2001 Nov 13 +2452227 +R +11 +3,4 +2.0213 +1.1434 +1.3073 +5.8479 +NV +NV +V +11 +3,4 +1.1840 +0.6858 +0.4703 +5.8479 +NV +NV +Note—S1 and S2 are the comparison and control star numbers respectively used for the IDV tests. Star +numbers follow the star maps shown in Table 5. + +AO 0235+164 optical variability +15 +Table 6. Result of scaled C-test and F-test for IDV on AO 0235+164 differential lightcurves from CASLEO +and CAHA (continued...) +Date +JD +Band +No. of +S1, S2 +Γ +CΓ +FΓ +F 0.005 +c +Status +Final +obs. +status +2005 Jan 16 +2453387 +R +11 +2,3 +1.5238 +3.8074 +14.4962 +5.8479 +V +V +2,4 +3.3465 +2.7388 +7.5010 +5.8479 +V +2,6 +3.3316 +2.7442 +7.5308 +5.8479 +V +2,7 +1.4051 +3.4058 +11.5996 +5.8479 +V +2005 Nov 2 +2453677 +R +32 +2,3 +1.2848 +6.4237 +41.2636 +2.5846 +V +V +2,4 +0.8959 +4.5227 +20.4545 +2.5846 +V +2,5 +0.2571 +4.3013 +18.5013 +2.5846 +V +2,6 +0.5453 +3.9310 +15.4528 +2.5846 +V +2,7 +0.5844 +4.9283 +24.2884 +2.5846 +V +2,8 +0.4738 +7.0560 +49.7865 +2.5846 +V +2,9 +0.3415 +6.5374 +42.7373 +2.5846 +V +2,10 +0.3397 +4.0828 +16.6695 +2.5846 +V +2005 Nov 4 +2453679 +R +12 +2,3 +0.8534 +4.3059 +18.5409 +5.3191 +V +V +2,4 +0.5599 +3.7978 +14.4235 +5.3191 +V +2,5 +0.1421 +5.0805 +25.8111 +5.3191 +V +2,6 +0.3029 +5.3341 +28.4524 +5.3191 +V +2,7 +0.3534 +7.6846 +59.0525 +5.3191 +V +2,8 +0.2839 +4.3153 +18.6220 +5.3191 +V +2,9 +0.1914 +11.0664 +122.4647 +5.3191 +V +2,10 +0.1875 +5.4019 +29.1804 +5.3191 +V +2005 Nov 5 +2453680 +R +44 +2,3 +0.9749 +10.6766 +113.9894 +2.2266 +V +V +2,4 +0.6398 +9.3431 +87.2939 +2.2266 +V +2,5 +0.1942 +11.0439 +121.9674 +2.2266 +V +2,6 +0.3721 +10.7338 +115.2142 +2.2266 +V +2,7 +0.4059 +10.2100 +104.2433 +2.2266 +V +2,8 +0.3427 +8.3494 +69.7127 +2.2266 +V +2,9 +0.2399 +12.1459 +147.5239 +2.2266 +V +2,10 +0.2340 +8.5775 +73.5744 +2.2341 +V +2005 Nov 6 +2453681 +R +40 +2,3 +1.0022 +7.8517 +61.6495 +2.3212 +V +V +2,4 +0.6946 +8.8524 +78.3645 +2.3212 +V +2,5 +0.2051 +6.6830 +44.6620 +2.3212 +V +2,6 +0.4022 +7.6630 +58.7211 +2.3212 +V +2,8 +0.3694 +5.9489 +35.3890 +2.3212 +V +2,9 +0.2576 +6.8684 +47.1751 +2.3212 +V +2,10 +0.2563 +5.1520 +26.5433 +2.3212 +V +Note—S1 and S2 are the comparison and control star numbers respectively used for the IDV tests. Star +numbers follow the star maps shown in Table 5. + +16 +Roy et al. +Table 6. Result of scaled C-test and F-test for IDV on AO 0235+164 differential lightcurves from CASLEO +and CAHA (continued...) +Date +JD +Band +No. of +S1, S2 +Γ +CΓ +FΓ +F 0.005 +c +Status +Final +obs. +status +2005 Nov 8 +2453683 +R +28 +2,3 +0.9329 +2.4256 +5.8834 +2.7940 +NV +NV +2,4 +0.6336 +2.3363 +5.4585 +2.7770 +NV +2,5 +0.1788 +1.7843 +3.1836 +2.7770 +NV +2,6 +0.3598 +2.2163 +4.9120 +2.7770 +NV +2,7 +0.4059 +1.4945 +2.2335 +2.7770 +NV +2,8 +0.3451 +2.1606 +4.6682 +2.9002 +NV +2,9 +0.2318 +1.3895 +1.9307 +2.7770 +NV +2005 Dec 5 +2453710 +R +20 +2,3 +1.4796 +1.4053 +1.9748 +3.4317 +NV +NV +2,4 +1.0247 +0.7240 +0.5242 +3.4317 +NV +2,5 +0.3133 +0.9355 +0.8752 +3.4317 +NV +2,6 +0.6030 +1.1896 +1.4151 +3.4317 +NV +2,8 +0.5634 +1.0332 +1.0674 +3.4317 +NV +2,9 +0.3979 +1.0716 +1.1482 +3.4317 +NV +2,10 +0.3915 +0.8994 +0.8089 +3.4317 +NV +2005 Dec 6 +2453711 +R +16 +2,3 +1.4092 +2.1709 +4.7129 +4.0698 +NV +PV +2,4 +0.9785 +3.7432 +14.0118 +4.0698 +V +2,5 +0.2848 +2.1323 +4.5467 +4.0698 +NV +2,6 +0.5570 +2.7562 +7.5967 +4.0698 +V +2,7 +0.6157 +1.8489 +3.4186 +4.0698 +NV +2,8 +0.5266 +1.3717 +1.8815 +4.0698 +NV +2,8 +0.5266 +1.3717 +1.8815 +4.0698 +NV +2,9 +0.3691 +2.4056 +5.7869 +4.0698 +NV +2,10 +0.3688 +2.0671 +4.2727 +4.0698 +NV +2019 Dec 17 +2458835 +R +30 +9,10 +1.3151 +1.2773 +1.6315 +2.6740 +NV +NV +9,11 +0.7377 +1.3155 +1.7307 +2.6740 +NV +9,12 +1.0425 +1.0698 +1.1445 +2.6740 +NV +Note—S1 and S2 are the comparison and control star numbers respectively used for the IDV tests. Star +numbers follow the star maps shown in Table 5. + +AO 0235+164 optical variability +17 +Table 7. Result of power enhanced F-test and nested ANOVA test for IDV on AO 0235+164 differential lightcurves from CASLEO and CAHA +Obs. +Band +No. of +Power enhanced F-test +Nested ANOVA test +Status +Variability +doubling +date +Obs. +Comp. +amplitude(%) +timescale +star +DOF(ν1,ν2) +Fenh +F 0.005 +c +DOF(ν1,ν2) +F +F 0.005 +c +(days) +1999 Nov 2 +V +23 +2 +(22, 87) +116.132 +2.209 +(5, 17) +58.924 +5.075 +V +43.99 +0.103 +1999 Nov 3 +V +22 +2 +(21, 42) +34.529 +2.540 +(5, 16) +10.920 +5.212 +V +24.47 +0.145 +1999 Nov 4 +V +30 +2 +(29, 58) +86.046 +2.216 +(7, 22) +38.922 +4.109 +V +34.48 +0.106 +R +30 +(29, 58) +82.016 +2.216 +(7, 22) +40.356 +4.109 +V +32.59 +0.083 +1999 Nov 5 +V +22 +2 +(21, 21) +9.207 +3.216 +(5, 16) +4.426 +5.212 +NV +10.94 +0.140 +R +23 +(22, 44) +2.951 +2.487 +(5, 17) +9.426 +5.075 +V +9.03 +0.335 +1999 Nov 6 +V +29 +2 +(28, 166) +211.363 +1.960 +(7, 21) +58.114 +4.179 +V +36.79 +0.092 +R +30 +(29, 170) +107.913 +1.941 +(7, 22) +74.686 +4.109 +V +37.90 +0.085 +1999 Nov 7 +V +12 +2 +(11, 55) +6.392 +2.854 +– +– +– +PV +9.13 +0.170 +R +11 +(10, 50) +6.413 +2.988 +– +– +– +PV +5.36 +0.244 +2000 Dec 21 +V +10 +2 +(9, 54) +4.813 +3.055 +– +– +– +PV +6.95 +0.275 +R +10 +(9, 54) +6.73 +3.055 +– +– +– +PV +7.67 +0.428 +2000 Dec 23 +V +11 +2 +(10, 70) +10.314 +2.846 +– +– +– +PV +20.58 +0.200 +R +10 +(9, 63) +14.542 +2.989 +– +– +– +PV +14.18 +0.180 +2001 Nov 9 +V +12 +2 +(11, 54) +2.345 +2.863 +– +– +– +NV +12.13 +0.372 +R +12 +(11, 22) +5.91 +3.612 +– +– +– +PV +12.73 +0.441 +2001 Nov 10 +V +10 +2 +(9, 27) +1.152 +3.557 +– +– +– +NV +8.49 +0.227 +R +10 +(9, 27) +1.054 +3.557 +– +– +– +NV +5.64 +0.660 +2001 Nov 11 +V +14 +2 +(13, 38) +2.02 +2.923 +– +– +– +NV +9.63 +0.364 +R +14 +(13, 38) +2.02 +2.923 +– +– +– +NV +9.63 +0.364 +2001 Nov 12 +V +12 +2 +(11, 44) +2.212 +2.969 +– +– +– +NV +12.06 +0.539 +R +12 +2 +(11, 22) +3.928 +3.612 +– +– +– +PV +11.74 +0.856 +2001 Nov 13 +V +11 +3 +(10, 10) +0.470 +5.847 +– +– +– +NV +10.81 +0.160 +R +11 +3 +(10, 10) +1.307 +5.847 +– +– +– +NV +10.19 +0.178 +2005 Jan 16 +R +11 +2 +(10, 40) +9.842 +3.117 +– +– +– +PV +32.92 +0.095 +2005 Nov 2 +R +32 +2 +(31, 247) +27.709 +1.868 +(7, 24) +37.156 +3.991 +V +8.98 +0.189 +2005 Nov 4 +R +12 +2 +(11, 88) +31.995 +2.689 +– +– +– +V +6.59 +0.166 +2005 Nov 5 +R +44 +2 +(43, 343) +124.459 +1.713 +(10, 33) +16.301 +3.26 +V +13.60 +0.146 +2005 Nov 6 +R +40 +2 +(39, 273) +57.755 +1.767 +(9, 30) +87.95 +3.45 +V +9.79 +0.227 +2005 Nov 8 +R +28 +2 +(27, 182) +4.371 +1.965 +(6, 21) +0.449 +4.393 +PV +3.18 +0.365 +2005 Dec 5 +R +20 +2 +(19, 133) +1.067 +2.200 +(4, 15) +14.394 +5.803 +PV +2.61 +0.391 +2005 Dec 6 +R +16 +2 +(15, 120) +4.863 +2.373 +– +– +– +PV +3.53 +0.746 +2019 Dec 17 +R +30 +9 +(29, 87) +1.453 +2.075 +(7, 22) +2.341 +4.109 +NV +7.74 +0.038 +Note—Comparison star numbers follow the star maps shown in Table 5. +to be affected only by instrumental fluctuations as any +known or suspected variable star can be discarded. +Jang & Miller (1997) and Romero et al. (1999) in- +troduced a parameter C defined as C = σB−S1/σS2−S1, +where σB−S1 and σS2−S1 are the standard deviations in +blazar DLC and control star DLC, respectively. +The +blazar is considered to be variable with 99.5 per cent +confidence level if C is greater than a critical value of +2.576. +Howell et al. (1988) pointed out that it is important +to select non-variable stars with magnitudes close to +the blazar magnitude as comparison and control stars. +Otherwise, even if the blazar is non-variable, there will +be difference between σB−S1 and σS2−S1 due to dif- + +18 +Roy et al. +Figure 9. Spectral fitting of AO 0235+164, where the black +line is the original spectrum while the green line is the single +power law for the fitted continuum. +The inset shows Mg +II line fitting where the blue, green, and red lines are the +narrow, broad, and total components, respectively. +ferences in photon statistics and other random-noise +terms (sky, read-out noise). To use field stars with dif- +ferent magnitude levels, Howell et al. (1988) suggests +calculating a correction factor Γ to scale σS2−S1 to the +instrumental level of σB−S1 for proper comparison. Γ +can be estimated using the following formula: +Γ2 = +�NS2 +NB +�2 � N 2 +S1(NB + P) + N 2 +B(NS1 + P) +N 2 +S2(NS1 + P) + N 2 +S1(NS2 + P) +� +(7) +where N is the total (sky-subtracted) counts within the +aperture, while the sub-indices B, S1 and S2 correspond +to N of the blazar, comparison star and control star, +respectively. The factor P contains the common noise- +terms, as P = npix(Nsky + N 2 +RON), where npix is the +number of pixels within the aperture, Nsky is the sky +level and NRON is the read-out noise. We used the me- +dian values of N of the objects and sky for calculating +Γ. Thus, the scaled C parameter (CΓ) is defined as +CΓ = C +Γ = 1 +Γ +� σB−S1 +σS2−S1 +� +. +(8) +The source is considered variable if CΓ ≥ 2.576. Even +though the C parameter is not a proper statistic, it re- +mains a useful indicator of stability (de Diego 2014; de +Diego et al. 2015; Zibecchi et al. 2017, 2020). +3.2.2. Scaled F-test +The +standard +F-statistics +parameter +is +F += +σ2 +B−S1/σ2 +S2−S1, where σ2 +B−S1 and σ2 +S2−S1 are the vari- +ances in blazar DLC and a control star DLC respectively. +The scaled F-statistics FΓ is given as +FΓ = F +Γ2 = 1 +Γ2 +� σ2 +B−S1 +σ2 +S2−S1 +� +. +The F-statistic assumes that the uncertainties in the +observations are normally distributed. If n(B−S1) and +n(S2−S1) are the sizes of the blazar and control star +DLC respectively, the number of degrees of freedom in +the numerator and denominator of the F-statistic are +ν1 = n(B−S1) − 1 and ν2 = n(S2−S1) − 1, respectively. +We calculated FΓ and considered the blazar to be vari- +able with 99.5 per cent confidence if FΓ was greater than +the critical value F α +c (ν1, ν2) at α = 0.005 (Zibecchi et al. +2017, 2020). +3.2.3. Power-enhanced F-test +The power-enhanced F -test (PEF) has been used in +various recent blazar IDV studies (Pandey et al. 2019; +Pandey et al. 2020, and references therein). The power- +enhanced F-statistic has the advantage of comparing the +blazar variance to the combined variance of multiple +field stars and is given as (de Diego 2014) +Fenh = s2 +blz +s2c +, +(9) +where s2 +blz is the variance of the DLC of the blazar with +respect to a reference star, and s2 +c is the combined vari- +ance of the comparison stars’ DLCs with respect to the +reference star. Thus, s2 +c is given as +s2 +c = +1 +��k +j=1 nj +� +− k +k +� +j=1 +nj +� +i=1 +s2 +j,i. +(10) +Here, k is the total number of available comparison stars +in the DLC, nj is the number of observations of the jth +comparison star, and s2 +j,i is the scaled square deviation +of the ith observation of the jth comparison star given +as +s2 +j,i = Γj(mj,i − ¯ +mj)2. +(11) +Here Γj is the scale factor of the jth comparison star +DLC computed following Equation 7. +Using the data of the field stars, we first checked the +star–star DLCs to identify any spikes due to instru- +mental errors or improper removal of cosmic rays, and +removed them iteratively if they were more than 3 +standard deviations from the mean magnitude. +We +considered a “well-behaved” star with low fluctuations +and an average magnitude close to the blazar as the +reference star. +The number of degrees of freedom in +the numerator and denominator of the F-statistics are + +AO 0235+164 optical variability +19 +ν1 = nblz − 1 and ν2 = +��k +j=1 nj +� +− k, respectively. +We calculated Fenh, and considered the blazar to be +variable (V) with 99.5 percent confidence if Fenh was +greater than the critical value Fc(ν1, ν2) at α = 0.005. +3.2.4. Nested ANOVA test +In the nested analysis of variance (ANOVA) test, DLCs +of the blazar are generated with respect to all the com- +parison stars used as reference stars. The details of this +method are given in de Diego et al. (2015). The nested +ANOVA test needs a large number of points in the light +curves, strongly limiting its application to densely pop- +ulated DLCs. We divided the DLCs with at least 20 +observations into groups such that each group contains +4 observations. Equation (4) of de Diego et al. (2015) +considers an ideal set of lightcurves where the total +number of observations are divisible by the group size. +In most of the DLCs in this work, the total number of +observations was not an integral multiple of the group +size of 4. So, in those cases, the last group contained less +than 4 observations, and we calculated the degrees of +freedom accordingly to compute the mean square due to +groups (MSG) and mean square due to the nested obser- +vations in groups (MSO(G)). The ANOVA F-statistic is +given as, F = MSG/MSO(G). For a significance level of +α = 0.005, if the F -statistic is greater than the critical +value (Fc), the blazar is taken as variable (V), other- +wise as non-variable (NV) with 99.5 per cent confidence. +We have listed the results of the scaled C-criterion +and scaled F-test in Table 6 and those of power en- +hanced F-test and the nested ANOVA test in Table 7. +In the case of scaled C-criterion and F-test, we fixed one +particular star as the comparison star for each dataset. +The source is declared variable with respect to one +comparison-control star pair if both scaled C-statistics +and F-statistics cross their respective critical values. +We declare the final variability status of the blazar +as variable/non-variable (V/NV) if it is variable/non- +variable against all control stars. If the blazar is variable +against some of the control stars, we call it probably +variable (PV). We did not carry out the nested ANOVA +test in a few datasets containing less than 20 obser- +vations. +In the case of the power-enhanced F-test in +absence of the corresponding nested ANOVA test, we +call the blazar probably variable (PV) even if the F- +statistic crosses the critical value, as the F-test is more +prone to give a false positive result (Zibecchi et al. 2017, +2020). If nested ANOVA is present and both the tests +cross the critical values, we call the blazar variable (V). +Otherwise, we declare the source non-variable (NV). We +list the summary of the IDV tests in Table 8. We give +a final verdict on the variability status of the source +after comparing the results of the combination of the +C-test and F-test (C&F) from Table 6 and results of +the combination of the power-enhance F-test and nested +ANOVA test (P&N) from Table 7. If the results from +both combinations were the same, we kept that result. +If C&F declared “V” and P&N declared “PV” due to +the absence of nested ANOVA, we finally consider the +source variable (V). We considered variability on 2005 +November 8 as “NV” because both C-test and nested +ANOVA resulted in non-variability. Despite being vari- +able in nested ANOVA, we consider the 2005 December +5 lightcurve “NV” as the F-test and PEF-test detected +no variability. A few examples of DLCs of AO 0235+164 +having different variability characteristics (V/PV/NV) +are shown in Figure 8. +3.2.5. Doubling timescale +A flux doubling/halving timescale gives an estimate of +the variability timescale (τvar) of a source. We calcu- +late the flux doubling/halving timescale (τd) between +two consecutive observations and its corresponding sig- +nificance (σ) as +F(ti+1) = F(ti) ∗ 2(ti+1−ti)/τd +σ = |F(ti+1) − F(ti)|/εi, +(12) +where F(ti) and εi are the flux observed at time ti +and the corresponding measurement uncertainty, respec- +tively. We consider the fastest doubling timescale (τ min +d +) +with a higher significance than 3σ as an estimate for +τvar. We obtained τ min +d +< 1 day for all the nights when +the source showed significant IDV both in scaled F-test +and nested ANOVA test. This further strengthens our +claims for the frequent presence of IDV. Following Equa- +tion 2 we computed the variability amplitudes on the +same nights. All these results are listed in Table 7. +3.2.6. Duty cycle +We calculated the duty cycle (DC) of AO 0235+164 +using the definition of Romero et al. (1999), that was +used later by multiple authors (e.g., Stalin et al. 2009; +Agarwal et al. 2016). The formula for DC for a partic- +ular waveband is given as, +DC = 100 +�n +i=1 Ni(1/∆ti) +�n +i=1(1/∆ti) % +(13) +where ∆ti = ∆ti,obs/(1+z) (duration of the monitoring +session on ith night is ∆ti,obs). Thus, this formula cal- +culates the duty cycle weighted by the cosmological red- +shift corrected monitoring duration of each night. We +set Ni = 1, 0.5, and 0 for the nights with variability + +20 +Roy et al. +Table 8. Summary of statistical tests for IDV on AO 0235+164 +differential lightcurves from CASLEO and CAHA +Obs. +Band +Combined variability status +Final +date +(C & F-test)a +(PEF & +status +N-ANOVA)b +1999 Nov 2 +V +V +V +V +1999 Nov 3 +V +V +V +V +1999 Nov 4 +V +V +V +V +R +V +V +V +1999 Nov 5 +V +V +NV +PV +R +NV +V +PV +1999 Nov 6 +V +V +V +V +R +V +V +V +1999 Nov 7 +V +PV +PV +PV +R +PV +PV +PV +2000 Dec 21 +V +PV +PV +PV +R +PV +PV +PV +2000 Dec 23 +V +PV +PV +PV +R +V +PV +V +2001 Nov 9 +V +NV +NV +NV +R +V +PV +V +2001 Nov 10 +V +NV +NV +NV +R +NV +NV +NV +2001 Nov 11 +V +NV +NV +NV +R +NV +NV +NV +2001 Nov 12 +V +NV +NV +NV +R +PV +PV +PV +2001 Nov 13 +V +NV +NV +NV +R +NV +NV +NV +2005 Jan 16 +R +V +PV +V +2005 Nov 2 +R +V +V +V +2005 Nov 4 +R +V +V +V +2005 Nov 5 +R +V +V +V +2005 Nov 6 +R +V +V +V +2005 Nov 8 +R +NV +PV +NV +2005 Dec 5 +R +NV +PV +NV +2005 Dec 6 +R +PV +PV +PV +2019 Dec 17 +R +NV +NV +NV +Note—aTable 6, bTable 7, PEF=power-enhanced F-test. +status “V”, “PV”, and “NV” respectively. We obtained +the duty cycle of AO 0235+164 to be ∼44 percent in V - +band, and ∼45 percent in R-band considering the nights +where the source was observed for at least 2 hours. +3.3. The mass of the central black hole +We estimate the mass of the SMBH in AO 0235+164 +by using its spectrum observed using the CCD Imag- +ing/Spectropolarimeter (SPOL) at the Steward Obser- +vatory4 on 2011 January 8 (air mass = 1.12). +This +spectrum was selected since the blazar was then at its +lowest level during the period 2008–2018, and should +ensure the best visibility of the emission lines because +of the lower continuum contribution from the jet. The +observed wavelength range of the spectrum we used is +4000–7550 ˚A, with a spectral resolution of 4 ˚A, and it is +analyzed by following the procedure given in Liao & Gu +(2020). Firstly, it was corrected for Galactic extinction +with the reddening map of Schlegel et al. (1998), and +then was shifted to the rest-frame wavelength by using +the redshift of 0.94. +This spectral coverage meant we could use the Mg +II line, which is prominent on the spectrum shown in +Figure 9 (focused on the 2400−3100 ˚A range), to es- +timate the SMBH mass. +We modeled the continuum +by applying a single power law (fλ ∝ λα) (as Fe II +emission is rather weak). A Gaussian profile was then +used to fit the Mg II line, centered at the position of +2800 ˚A, on the continuum-subtracted spectrum. +The +broad component of Mg II was fitted with a Gaussian +with a 1000 km s−1 lower limit, while a Gaussian with +upper limit of 1000 km s−1 was applied for the narrow +component. +In order to estimate the corresponding +errors of full width at half maximum (FWHM) and +flux, we generated 100 mock spectra by adding random +Gaussian noise to the original spectrum using the flux +density errors, and then took the standard deviation of +measurements from those mock spectra as the uncer- +tainties. +Here, the flux density errors were the RMS +value of the spectrum calculated over the spectral win- +dow of (3000−3100) ˚A, after subtracting a second-order +polynomial function. +Figure 9 shows the resulting fit +to the spectrum. Our best fitting results indicate that +the line width of the broad Mg II component is FWHM += 3151 km s−1, with log-scale luminosity in erg s−1, +log(LMgII) = 42.8. +The line width and the Mg II line luminosity we find +are consistent with the range of values FWHM=3100– +3500 km s−1 and log(LMgII)=42.5–42.8, respectively, +which were derived by Raiteri et al. (2007) from one +VLT and four TNG spectra of AO 0235+164 acquired +in 2003–2004. +We use the FWHM and luminosity of +the broad Mg II line, not the continuum luminosity, as +4 http://james.as.arizona.edu/∼psmith/Fermi + +AO 0235+164 optical variability +21 +1015 +1016 + (HZ) +10 +14 +10 +13 +10 +12 +F (erg cm +2 s +1) +Disk thermal +U +B +V +R +I +JD 2452169 +Figure 10. Comparison of the SED of the lowest flux state +observed on JD 2452169 and the thermal emission from the +accretion disk in the observer’s frame. The thermal emis- +sion component is calculated using a multi-temperature disk +model with the black hole mass log(MBH/M⊙) = 7.9±0.25, +and the log-scale disk luminosity in erg s−1, log(Ldisk) = +45.01±0.20. The shaded region indicates the uncertainties +in the calculation of the disk thermal component. +we are unable to exclude the jet emission contribution, +despite the low state spectrum that we could use for +this blazar. +The black hole mass is derived from the +empirical relation used for Mg II (Kong et al. 2006), +which is based on measured broad line region sizes in +the reverberation-mapping AGN sample of Peterson +et al. (2004), as +MBH +M⊙ += 2.9×106 +� +LMgII +1042 erg s−1 +�0.57±0.12 �FWHMMgII +103 km s−1 +�2 +(14) +Thus, the SMBH mass is log(MBH/M⊙) = 7.90 ± 0.25, +where the uncertainty is estimated from the measure- +ment uncertainties of the FWHM and luminosity of +Mg II. Using optical spectroscopy data from the SDSS +archive, Paliya et al. (2021) reported a somewhat higher +mass, log(MBH/M⊙) = 8.58 ± 0.34, and an accretion +disk luminosity (in erg s−1), of log(Ldisk) = 45.30 ± +0.22. Using the method mentioned in Paliya et al. (2021) +with log(LMgII) = 42.8, we obtained a lower disk lumi- +nosity (in erg s−1) of log(Ldisk) = 45.01 ± 0.20 from the +spectrum observed on 2011 January 8. +4. DISCUSSION +In this work, we have presented a detailed temporal +and spectral study of the highly variable emission from +the blazar AO 0235+164 observed at multiple optical +wavebands (UBVRI) from October 1975 to December +2019. The lightcurves have highly uneven data sampling +due to gaps in observation seasons and non-uniform ob- +servation campaigns. Although U-band data are quite +sparsely sampled the BVRI observations have denser +sampling when the source was highly active. Multiple +long-term studies suggested that AO 0235+164 shows +∼2-year long flaring episodes with multiple sub-flares +after intervals of ∼8 years (Raiteri et al. 2006; Fan et al. +2017; Roy et al. 2022). Figure 1 shows a difference of +about six magnitudes between the quiescent and out- +burst states in all optical wavebands, corresponding to +an energy flux variation of more than two orders of +magnitude (Figure 6). The long-term variability ampli- +tudes at all five wavebands are quite similar (Table 1). +Also, we found a strong correlation with zero time-lag +between the UBVI observations and the R-band data +(Figure 2 and Figure 3), which implies a common ra- +diative process at a single emission zone is responsible +for the bulk of the emission at the optical wavebands. +Sometimes during the quiescent states of powerful +blazars, the disk thermal emission component becomes +visible as a big blue bump on top of the synchrotron +emission component from the jet in the optical-UV +wavebands (e.g., Roy et al. 2021). As the disk emission +is bluer than the jet synchrotron emission, an increase +in the jet activity during low flux states displays a +redder-when-brighter trend. +The enhanced jet activ- +ity is observed when the charged particles inside the +jet get accelerated to higher energies, and then radiate +faster. Thus, the jet synchrotron component tends to +get bluer with the increase in flux. If the jet emission +completely outshines the disk emission, we expect to +see a bluer-when-brighter trend (e.g., Isler et al. 2017). +The flux increment can also be attributed to the in- +crease in the jet Doppler factor (e.g., Papadakis et al. +2007), which blueshifts the spectrum and produces a +bluer-when-brighter trend because of the convexity of +the spectrum. Such a trend is seen in the (B − I) vs R +magnitude diagram (Figure 4b) and indicates the dom- +ination of non-thermal jet emission over the thermal +emission component of the accretion disk during both +flaring and quiescent states. +From the convex shapes +of the optical BVR SEDs during states ranging from +quiescent to flaring (see the accompanying SED video +and Figure 6), we may infer that the effect of the disk +thermal emission is not significant in optical wavebands +even during the low flux states. +This can be explained in terms of the nature of disk +thermal emission given the disk luminosity and the cen- +tral black hole mass computed in subsection 3.3. The +primary, and most precise, black hole mass estimation +methods are based on stellar and gas kinematics and +reverberation mapping (e.g. Vestergaard 2004). These + +22 +Roy et al. +methods need high spatial resolution spectroscopy data +from the host galaxy and/or higher ionization emission +lines and are not applicable to most BL Lacertae objects +(BL Lacs). But in BL Lacs, if the weak emission lines +are present, we can use the empirical methods (Kong +et al. 2006) for BH mass estimation. The most common +methods used for BH mass estimation for BL Lacs are +the shortest variability timescales and periods of QPOs +(Gupta et al. 2012). Since BL Lacs are highly variable +objects, any BH mass estimation may be treated as an +upper limit, and there are possibilities of detection of +a shorter variability timescale or shorter QPO period. +We obtained a log-scale BH mass of 7.90±0.25 in so- +lar mass unit. The Steward observatory spectrum we +used in our analysis had a narrower Mg II emission +line (FWHM=3151 km s−1) than those of Raiteri et al. +(2007) and Paliya et al. (2021), thus resulting in a lower +mass estimate. +We considered a multi-temperature +blackbody type accretion disk model, where the temper- +ature at any portion of the disk is a function of the disk +luminosity and the central black hole mass, to compute +the thermal emission component. In Figure 10 we plot- +ted the thermal component along with the optical-UV +SED during the lowest activity state of AO 0235+164 +observed on JD 2452169. It is evident that, as the ther- +mal emission peaks at far UV frequencies (∼3.5×1015 +Hz) in the observer’s frame of reference, the jet emission +always dominates in BVRI wavebands. We do not see +any significant trend in the variation of the (V − R) +spectral index (αV R) (Figure 7). +The sudden rise of +the U-band flux in Figure 10 is an indicator of a prob- +able UV-soft X-ray bump as discussed in Raiteri et al. +(2005, 2006). +According to these studies, the source +of the bump is either an additional synchrotron com- +ponent coming from a separate emission region in the +jet or the emission of a continuous inhomogeneous jet +is suppressed in near UV region due to a discontinuity +in opacity or misalignment of that particular emission +region. +Ackermann et al. (2012) mentioned that the +whole optical-UV spectrum is produced by a single syn- +chrotron emitting zone as the shape of the bump does +not change with luminosity. +They attributed the UV +spectral hardening to an artifact due to the overestima- +tion of extinction by Junkkarinen et al. (2004). +For the detection of any statistically significant intraday +variability in 33 lightcurves of AO 0235+164 observed +at CASLEO/CAHA, we employed different statistical +tests widely used in AGN variability studies. The re- +liability of each of these tests has been disputed (e.g. +de Diego et al. 2015; Zibecchi et al. 2017), so we here +employed a comparative approach that could allow us +to circumvent the limitations affecting any individual +test. In the first place, we used the scaled C-criterion +and the F-test. The first compares the dispersion of the +blazar lightcurve to the dispersion of a field star (con- +trol star), while the latter does so with the variances. +According to Zibecchi et al. (2017) and Zibecchi et al. +(2020), the F-test has a tendency to classify noisy non- +variable curves as a variable (i.e., give false positives), +while the C-criterion tends to give false negatives. Even +though the C-criterion (Romero et al. 1999) cannot be +considered as an actual statistical test, it may still be a +useful parameter to detect variability with high signifi- +cance. The F-test, on the other hand, does not always +work as expected, because it is particularly sensitive to +non-Gaussian errors (“red noise”), which are usually an +issue when analyzing blazars DLCs. +We also used the power-enhanced F-test and the nested +ANOVA test, which involve multiple field stars. It is ex- +pected that the power-enhanced F-test may also suffer +from the same drawback of detecting false variability +as the (original) F-test. +In the nested ANOVA test, +in turn, data grouping may lead to false results if data +within a time span larger than the (unknown) variability +timescale are grouped. Comparing the results of Table 6 +and Table 7, while considering the tendencies of giving +false results by the respective tests, we can confirm that +the source was significantly variable in 4 out of 13 V - +band lightcurves, and 9 out of 20 R-band lightcurves. +The source seems to be probably variable in 3 V -band +and 4 R-band lightcurves, and non-variable in the rest. +On 1999 November 5, the combination of C-criterion +and F-test indicates non-variability but the combination +of power-enhanced F-test and nested ANOVA detects +variability in the R-band lightcurve. The results in the +V -band lightcurve on that day are exactly the opposite. +Similar situations were observed also on 2001 November +9 and 2001 November 12. +A visual inspection of the +DLCs of these nights reveals that the blazar DLCs were +classified as non-variable when either the control star +DLC had higher variability (1999 November 5) or the +measurement errors of the blazar DLCs were higher due +to its low-flux state (2001 November 9 and 12). Higher +measurement errors lead to a lower chance of signifi- +cant variability detection. +These strange results may +be an example of the drawbacks of the applied methods +when trying to recover low-amplitude variations from +DLCs affected by non-Gaussian noise (part of the ob- +servations on that night were taken at air mass > 2 +and under non-photometric conditions). Otherwise, the +combined results of different methods seem to more or +less agree. +Alongside the optical SED patterns, such +frequent IDV establishes AO 0235+164 as a low-energy + +AO 0235+164 optical variability +23 +Table 9. +Variation of duty cycle with +the duration of observation in R-band. +Observation +No. of +Duty +duration (hours) +nights +cycle (%) +> 1 +20 +52 +> 2 +19 +45 +> 3 +17 +50 +> 4 +14 +57 +> 5 +13 +64 +> 6 +8 +77 +peaked BL Lac (LBL) object. High energy peaked BL +Lacs (HBL) show significantly less optical intraday vari- +ability than the LBLs (Heidt & Wagner 1998; Romero +et al. 1999). +The differences in IDV behavior have been attributed +to the strength of magnetic fields present in the jet of +HBLs. A higher axial magnetic field (B) than a critical +value (Bc) may prevent the generation of any bends and +Kelvin-Helmhotz instabilities in the jet-base responsible +for creating intraday microvariabilities. This indicates +the presence of a weaker magnetic field than Bc in the +jet of AO 0235+164. The critical magnetic field (Bc) is +given in Romero (1995) as +Bc = +� +4πnemec2(Γ2 − 1)/Γ, +(15) +where ne is the electron density in the emission region, +me is the electron rest mass, and here Γ is the bulk +Lorentz factor of the jet flow. Considering a typical set +of parameters, ne = 429 cm−3 and Γ = 20 (Ackermann +et al. 2012), we get Bc ≃ 0.07 G. +From Table 7 and Figure 8, we can say that the vari- +ability amplitudes were higher in the 1999 season when +the source was in a fainter state (higher magnitude) than +its brighter state in the 2005 season. Marscher (2013) +suggested that enhancement of flux can arise from a +more uniform flow of particles inside the jet, which in +turn decreases the amplitude of microvariability asso- +ciated with the turbulence inside the jet. Equation 9 +indicates that the probability of detection of significant +variability increases with the duration of observation. +Similar results for other blazars were found by Gupta +& Joshi (2005), Rani et al. (2010), and Agarwal et al. +(2016). +From the flux doubling timescales listed in Table 7, we +can estimate the upper limit to the size of the emission +region (Rmax) using the light travel-time argument given +as +Rmax = cδtvar +1 + z +(16) +where z is the cosmological redshift of 0.94, tvar is the +variability timescale, and δ is the Doppler boost of the +jet. Considering δ = 24 (Hovatta et al. 2009) and tvar +to be the shortest flux doubling timescale of 0.083 days +(when the source was significantly variable), we obtain +an emission region size upper limit of ∼ 2.6 × 1015 cm. +Assuming a conical jet model where the emission re- +gion fills up the entire jet cross-section, we can estimate +the probable maximum distance (dmax) of the emission +region from the central black hole as, dmax = ΓRmax = +5.2×1016 cm. To explain the observed strong variability, +Marchesini et al. (2016) attempted to apply a swinging +jet model that attributes the observed variability to a +change in the viewing angle of the emission region with +time (i.e. variation in the associated bulk Doppler fac- +tor). They reported a high rate of change in viewing an- +gle of about 7−10 arcmin per day, considering a mean +viewing angle of 2.3◦, would be necessary. +However, +they found that this geometric wiggling-jet scenario was +disfavored when considering the observed variation in +color index with time. +Several earlier studies on AO +0235+164 associated the observed fast optical variabil- +ity with gravitational microlensing by the foreground +absorber at z = 0.524. Webb et al. (2000) proposed that +the 1997 flare resulted due to microlensing because of an +observed correlation with zero lag between radio and op- +tical lightcurves following Stickel et al. (1988), but the +absence of any correlated flare in the X-ray lightcurve +makes this explanation less likely. Abraham et al. (1993) +and Raiteri et al. (2007) explained that such microlens- +ing events can produce small amounts of fast flux ampli- +fication but are unlikely to dominate the high variability +observed in AO 0235+164. +5. CONCLUSIONS +In this work, we conducted a study of long-term and +short-term (intraday) variability in the optical mul- +tiwaveband observations of the blazar AO 0235+164. +Here we summarize our results and the probable physi- +cal scenarios. +1. We observed a variation of about six magnitudes +between the quiescent and flaring episodes, or over +two orders of magnitude variation in the SEDs. +2. UBVI lightcurves are highly correlated with the +R-band lightcurve with zero time lag. + +24 +Roy et al. +3. A significant bluer-when-brighter trend is observed +in the (B − I) color variation with R-magnitude. +4. All the optical BVR-band SEDs show convexity. +These observations indicate that the optical emis- +sion is dominated by jet radiation. +5. AO 0235+164 frequently shows statistically sig- +nificant intraday variability in optical wavebands. +This implies that AO 0235+164 is an LBL and +probably has a weak magnetic field in the jet en- +vironment. +6. From the analysis of a broad Mg II emission line +in a spectrum of AO 0235+164 taken at a low +state, we estimate a central black-hole mass of ∼ +7.9 × 107M⊙. +ACKNOWLEDGMENTS +Data from the Steward Observatory spectropolari- +metric monitoring project were used. +This pro- +gram is supported by Fermi Guest Investigator grants +NNX08AW56G, NNX09AU10G, NNX12AO93G, and +NNX15AU81G. This paper has made use of up-to- +date SMARTS optical/near-infrared light curves that +are available at www.astro.yale.edu/smarts/glast/home. +php. This work is partly based on data taken and as- +sembled by the WEBT collaboration and stored in the +WEBT archive at the Osservatorio Astrofisico di Torino +- +INAF +(https://www.oato.inaf.it/blazars/webt/). +These data are available upon request to the WEBT +President Massimo Villata (massimo.villata@inaf.it). +This work is based on data acquired at Complejo +Astron´omico El Leoncito, operated under an agree- +ment between the Consejo Nacional de Investigaciones +Cient´ıficas y T´ecnicas de la Rep´ublica Argentina and +the National Universities of La Plata, C´ordoba and San +Juan. We thank Anabella Araudo and Ileana Andru- +chow for help with the observations made with CASLEO +and the data analysis. +We thankfully acknowledge the anonymous reviewer +for very useful comments which helped us to improve +the manuscript. +We acknowledge the support of the +Department of Atomic Energy, Government of India, +under project identification number RTI 4002. +ACG +is partially supported by Chinese Academy of Sciences +(CAS) President’s International Fellowship Initiative +(PIFI) (grant no. +2016VMB073). +GER acknowl- +edges support from grants PIP 0554 (CONICET), +PIP +2021-1639 +(CONICET), +and +grant +PID2019- +105510GBC31 of the Spanish Ministerio de Ciencia, +Innovaci´on y Universidades and through the Center +of Excellence Mara de Maeztu 2020-2023 award to +the ICCUB (CEX2019-000918-M). JAC is Mar´ıa Zam- +brano researcher fellow funded by the European Union +-NextGenerationEU- (UJAR02MZ), supported by PIP +0113 (CONICET) and PICT-2017-2865 (ANPCyT). +JAC was also supported by grant PID2019-105510GB- +C32/AEI/10.13039/501100011033 from the Agencia Es- +tatal de Investigaci´on of the Spanish Ministerio de +Ciencia, Innovaci´on y Universidades, and by Consejer´ıa +de Econom´ıa, Innovaci´on, Ciencia y Empleo of Junta +de Andaluc´ıa as research group FQM-322, as well as +FEDER funds. +Facilities: +WEBT, SMARTS, Bok, +SO:Kuiper, +MMT, CASLEO:JST, CAO:2.2m + +AO 0235+164 optical variability +25 +Software: +Astropy (Astropy Collaboration et al. +2013), DAOPHOT (Stetson 1987), IRAF (Tody 1986) +REFERENCES +Abraham, R. G., Crawford, C. S., Merrifield, M. R., +Hutchings, J. B., & McHardy, I. M. 1993, ApJ, 415, 101, +doi: 10.1086/173147 +Ackermann, M., Ajello, M., Ballet, J., et al. 2012, ApJ, 751, +159, doi: 10.1088/0004-637X/751/2/159 +Ackermann, M., Ajello, M., Ballet, J., et al. 2012, ApJ, 751, +doi: 10.1088/0004-637X/751/2/159 +Agarwal, A., Gupta, A. C., Bachev, R., et al. 2016, +MNRAS, 455, 680, doi: 10.1093/mnras/stv2345 +Agudo, I., Marscher, A. P., Jorstad, S. G., et al. 2011, +ApJL, 735, L10, doi: 10.1088/2041-8205/735/1/L10 +Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., +et al. 2013, A&A, 558, A33, +doi: 10.1051/0004-6361/201322068 +Bessell, M. S. 2005, ARA&A, 43, 293, +doi: 10.1146/annurev.astro.41.082801.100251 +Bonning, E., Urry, C. M., Bailyn, C., et al. 2012, The +Astrophysical Journal, 756, 13, +doi: 10.1088/0004-637X/756/1/13 +Cellone, S. A., Romero, G. E., Combi, J. A., & Mart´ı, J. +2007, MNRAS, 381, L60, +doi: 10.1111/j.1745-3933.2007.00366.x +Cohen, R. D., Smith, H. E., Junkkarinen, V. T., & +Burbidge, E. M. 1987, ApJ, 318, 577, doi: 10.1086/165393 +de Diego, J. A. 2014, AJ, 148, 93, +doi: 10.1088/0004-6256/148/5/93 +de Diego, J. A., Polednikova, J., Bongiovanni, A., et al. +2015, AJ, 150, 44, doi: 10.1088/0004-6256/150/2/44 +Edelson, R. A., & Krolik, J. H. 1988, ApJ, 333, 646, +doi: 10.1086/166773 +Fan, J. H., & Lin, R. G. 1999, ApJS, 121, 131, +doi: 10.1086/313191 +Fan, J. H., Tao, J., Qian, B. C., et al. 2006, Publications of +the Astronomical Society of Japan, 58, 797, +doi: 10.1093/pasj/58.5.797 +Fan, J. H., Kurtanidze, O., Liu, Y., et al. 2017, ApJ, 837, +45, doi: 10.3847/1538-4357/aa5def +Fossati, G., Maraschi, L., Celotti, A., Comastri, A., & +Ghisellini, G. 1998, MNRAS, 299, 433, +doi: 10.1046/j.1365-8711.1998.01828.x +Gonz´alez-P´erez, J. N., Kidger, M. R., & Mart´ın-Luis, F. +2001, AJ, 122, 2055, doi: 10.1086/322129 +Guo, Y. C., Hu, S. M., Xu, C., et al. 2015, NewA, 36, 9, +doi: 10.1016/j.newast.2014.09.011 +Gupta, A. C., Banerjee, D. P. K., Ashok, N. M., & Joshi, +U. C. 2004, A&A, 422, 505, +doi: 10.1051/0004-6361:20040306 +Gupta, A. C., Fan, J. H., Bai, J. M., & Wagner, S. J. 2008, +AJ, 135, 1384, doi: 10.1088/0004-6256/135/4/1384 +Gupta, A. C., & Joshi, U. C. 2005, A&A, 440, 855, +doi: 10.1051/0004-6361:20042370 +Gupta, S. P., Pandey, U. S., Singh, K., et al. 2012, NewA, +17, 8, doi: 10.1016/j.newast.2011.05.005 +Hagen-Thorn, V. A., Larionov, V. M., Jorstad, S. G., et al. +2008, ApJ, 672, 40, doi: 10.1086/523841 +Heidt, J., & Wagner, S. J. 1996, A&A, 305, 42. +https://arxiv.org/abs/astro-ph/9506032 +—. 1998, A&A, 329, 853. +https://arxiv.org/abs/astro-ph/9709116 +Hovatta, T., Valtaoja, E., Tornikoski, M., & L¨ahteenm¨aki, +A. 2009, A&A, 494, 527, +doi: 10.1051/0004-6361:200811150 +Howell, S. B., Mitchell, K. J., & Warnock, A. I. 1988, AJ, +95, 247 +Ikejiri, Y., Uemura, M., Sasada, M., et al. 2011, PASJ, 63, +639, doi: 10.1093/pasj/63.3.327 +Impey, C. D., Brand, P. W. J. L., & Tapia, S. 1982, +MNRAS, 198, 1, doi: 10.1093/mnras/198.1.1 +Isler, J. C., Urry, C. M., Coppi, P., et al. 2017, The +Astrophysical Journal, 844, 107, +doi: 10.3847/1538-4357/aa79fc +Itoh, R., Nalewajko, K., Fukazawa, Y., et al. 2016, ApJ, +833, 77, doi: 10.3847/1538-4357/833/1/77 +Jang, M., & Miller, H. R. 1997, AJ, 114, 565, +doi: 10.1086/118493 +Junkkarinen, V. T., Cohen, R. D., Beaver, E. A., et al. +2004, ApJ, 614, 658, doi: 10.1086/423777 +Kong, M.-Z., Wu, X.-B., Wang, R., & Han, J.-L. 2006, +ChJA&A, 6, 396, doi: 10.1088/1009-9271/6/4/02 +Kutkin, A. M., Pashchenko, I. N., Lisakov, M. M., et al. +2018, MNRAS, 475, 4994, doi: 10.1093/mnras/sty144 +Landolt, A. U. 2009, AJ, 137, 4186, +doi: 10.1088/0004-6256/137/5/4186 +Liao, M., & Gu, M. 2020, MNRAS, 491, 92, +doi: 10.1093/mnras/stz2981 +Madejski, G., Takahashi, T., Tashiro, M., et al. 1996, ApJ, +459, 156, doi: 10.1086/176877 +Marchesini, E. J., Andruchow, I., Cellone, S. A., et al. 2016, +A&A, 591, A21, doi: 10.1051/0004-6361/201527632 + +26 +Roy et al. +Marscher, A. P. 1983, ApJ, 264, 296, doi: 10.1086/160597 +Marscher, A. P. 2013, The Astrophysical Journal, 780, 87, +doi: 10.1088/0004-637x/780/1/87 +Miller, H. R., Carini, M. T., & Goodrich, B. D. 1989, +Nature, 337, 627, doi: 10.1038/337627a0 +M¨ucke, A., Protheroe, R. J., Engel, R., Rachen, J. P., & +Stanev, T. 2003, Astroparticle Physics, 18, 593, +doi: 10.1016/S0927-6505(02)00185-8 +Nilsson, K., Charles, P. A., Pursimo, T., et al. 1996, A&A, +314, 754 +Paliya, V. S., Dom´ınguez, A., Ajello, M., Olmo-Garc´ıa, A., +& Hartmann, D. 2021, ApJS, 253, 46, +doi: 10.3847/1538-4365/abe135 +Pandey, A., Gupta, A. C., Wiita, P. J., & Tiwari, S. N. +2019, ApJ, 871, 192, doi: 10.3847/1538-4357/aaf974 +Pandey, A., Gupta, A. C., Kurtanidze, S. O., et al. 2020, +The Astrophysical Journal, 890, 72, +doi: 10.3847/1538-4357/ab698e +Papadakis, I. E., Villata, M., & Raiteri, C. M. 2007, A&A, +470, 857, doi: 10.1051/0004-6361:20077516 +Peterson, B. M., Ferrarese, L., Gilbert, K. M., et al. 2004, +ApJ, 613, 682, doi: 10.1086/423269 +Qian, S. J., Kraus, A., Witzel, A., Krichbaum, T. P., & +Zensus, J. A. 2000, A&A, 357, 84 +Rabbette, M., McBreen, S., Steel, B., & Smith, N. 1996, +A&A, 310, 1 +Raiteri, C. M., Villata, M., Capetti, A., et al. 2007, A&A, +464, 871, doi: 10.1051/0004-6361:20066599 +Raiteri, C. M., Villata, M., Aller, H. D., et al. 2001, A&A, +377, 396, doi: 10.1051/0004-6361:20011112 +Raiteri, C. M., Villata, M., Ibrahimov, M. A., et al. 2005, +A&A, 438, 39, doi: 10.1051/0004-6361:20042567 +Raiteri, C. M., Villata, M., Kadler, M., et al. 2006, A&A, +459, 731, doi: 10.1051/0004-6361:20065744 +Raiteri, C. M., Villata, M., Larionov, V. M., et al. 2008, +A&A, 480, 339, doi: 10.1051/0004-6361:20079044 +Raiteri, C. M., Villata, M., Acosta-Pulido, J. A., et al. +2017, Nature, 552, 374, doi: 10.1038/nature24623 +Rani, B., Gupta, A. C., Strigachev, A., et al. 2010, Monthly +Notices of the Royal Astronomical Society, 404, 1992, +doi: 10.1111/j.1365-2966.2010.16419.x +Romero, G. E. 1995, Ap&SS, 234, 49, +doi: 10.1007/BF00627281 +Romero, G. E., Boettcher, M., Markoff, S., & Tavecchio, F. +2017, SSRv, 207, 5, doi: 10.1007/s11214-016-0328-2 +Romero, G. E., Cellone, S. A., & Combi, J. A. 1999, +A&AS, 135, 477, doi: 10.1051/aas:1999184 +—. 2000, A&A, 360, L47. +https://arxiv.org/abs/astro-ph/0007407 +Romero, G. E., Cellone, S. A., Combi, J. A., & Andruchow, +I. 2002, A&A, 390, 431, doi: 10.1051/0004-6361:20020743 +Roy, A., Patel, S. R., Sarkar, A., Chatterjee, A., & Chitnis, +V. R. 2021, MNRAS, 504, 1103, +doi: 10.1093/mnras/stab975 +Roy, A., Chitnis, V. R., Gupta, A. C., et al. 2022, MNRAS, +513, 5238, doi: 10.1093/mnras/stac1287 +Sagar, R., Stalin, C. S., Gopal-Krishna, & Wiita, P. J. 2004, +MNRAS, 348, 176, doi: 10.1111/j.1365-2966.2004.07339.x +Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, +500, 525, doi: 10.1086/305772 +Schramm, K. J., Borgeest, U., Kuehl, D., et al. 1994, +A&AS, 106, 349 +Smith, P. S., Balonek, T. J., Heckert, P. A., Elston, R., & +Schmidt, G. D. 1985, AJ, 90, 1184, doi: 10.1086/113824 +Smith, P. S., Montiel, E., Rightley, S., et al. 2009, arXiv +e-prints, arXiv:0912.3621. +https://arxiv.org/abs/0912.3621 +Stalin, C. S., Kawabata, K. S., Uemura, M., et al. 2009, +Monthly Notices of the Royal Astronomical Society, 399, +1357, doi: 10.1111/j.1365-2966.2009.15354.x +Stetson, P. B. 1987, Publications of the Astronomical +Society of the Pacific, 99, 191, doi: 10.1086/131977 +Stickel, M., Fried, J. W., & Kuehr, H. 1988, A&A, 198, L13 +—. 1993, A&AS, 98, 393 +Takalo, L. O., Sillanpaeae, A., Valtaoja, E., et al. 1998, +A&AS, 129, 577, doi: 10.1051/aas:1998205 +Tody, D. 1986, in Society of Photo-Optical Instrumentation +Engineers (SPIE) Conference Series, Vol. 627, +Instrumentation in astronomy VI, ed. D. L. Crawford, +733, doi: 10.1117/12.968154 +Urry, C. M., & Padovani, P. 1995, PASP, 107, 803, +doi: 10.1086/133630 +Vestergaard, M. 2004, in Astronomical Society of the +Pacific Conference Series, Vol. 311, AGN Physics with +the Sloan Digital Sky Survey, ed. G. T. Richards & P. B. +Hall, 69. https://arxiv.org/abs/astro-ph/0401436 +Villata, M., Raiteri, C. M., Kurtanidze, O. M., et al. 2002, +A&A, 390, 407, doi: 10.1051/0004-6361:20020662 +Villata, M., Raiteri, C. M., Larionov, V. M., et al. 2008, +A&A, 481, L79, doi: 10.1051/0004-6361:200809552 +Villata, M., Raiteri, C. M., Gurwell, M. A., et al. 2009, +A&A, 504, L9, doi: 10.1051/0004-6361/200912732 +Wagner, S. J., & Witzel, A. 1995, ARA&A, 33, 163, +doi: 10.1146/annurev.aa.33.090195.001115 +Wang, Y.-F., & Jiang, Y.-G. 2020, ApJ, 902, 41, +doi: 10.3847/1538-4357/abb36c +Webb, J. R., Howard, E., Ben´ıtez, E., et al. 2000, AJ, 120, +41, doi: 10.1086/301432 + +AO 0235+164 optical variability +27 +White, R. J., & Peterson, B. M. 1994, PASP, 106, 879, +doi: 10.1086/133456 +Wierzcholska, A., Ostrowski, M., Stawarz, �L., Wagner, S., +& Hauser, M. 2015, A&A, 573, A69, +doi: 10.1051/0004-6361/201423967 +Woo, J.-H., & Urry, C. M. 2002, ApJ, 579, 530, +doi: 10.1086/342878 +Zhang, B.-K., Jin, M., Zhao, X.-Y., Zhang, L., & Dai, B.-Z. +2021, Research in Astronomy and Astrophysics, 21, 186, +doi: 10.1088/1674-4527/21/8/186 +Zibecchi, L., Andruchow, I., Cellone, S. A., & Carpintero, +D. D. 2020, MNRAS, 498, 3013, +doi: 10.1093/mnras/staa2544 +Zibecchi, L., Andruchow, I., Cellone, S. A., et al. 2017, +MNRAS, 467, 340, doi: 10.1093/mnras/stx054 + diff --git a/99AzT4oBgHgl3EQf_P4t/content/tmp_files/load_file.txt b/99AzT4oBgHgl3EQf_P4t/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a8c2943f6d281a929573b8ff0b476f15ba46b4d2 --- /dev/null +++ b/99AzT4oBgHgl3EQf_P4t/content/tmp_files/load_file.txt @@ -0,0 +1,2704 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf,len=2703 +page_content='Draft version January 6, 2023 Typeset using LATEX twocolumn style in AASTeX63 Study of variability in long-term multiwavelength optical lightcurves of blazar AO 0235+164 Abhradeep Roy ,1 Alok C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Gupta ,2, 3 Varsha R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Chitnis ,1 Sergio A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Cellone ,4, 5 Claudia M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Raiteri ,6 Gustavo E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Romero ,7, 5 Paul J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Wiita ,8 Anshu Chatterjee ,1 Jorge A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Combi ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 9 Mai Liao ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 11 Arkadipta Sarkar ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='12 and Massimo Villata 6 1Department of High Energy Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Tata Institute of Fundamental Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Homi Bhabha Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Mumbai-400005,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' India 2Aryabhatta Research Institute of Observational Sciences (ARIES),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Manora Peak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Nainital 263001,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' India 3Key Laboratory for Research in Galaxies and Cosmology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Shanghai Astronomical Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Shanghai 200030,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' China 4Complejo Astron´omico El Leoncito (CASLEO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' CONICET-UNLP-UNC-UNSJ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' San Juan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Argentina 5Facultad de Ciencias Astron´omicas y Geof´ısicas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Universidad Nacional de La Plata,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' La Plata,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Buenos Aires,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Argentina 6INAF-Osservatorio Astrofisico di Torino,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Via Osservatorio 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' I-10025 Pino Torinese,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Italy 7Instituto Argentino de Radioastronom´ıa (CCT-La Plata,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' CONICET;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' CICPBA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' UNLP), Buenos Aires, Argentina 8Department of Physics, The College of New Jersey, 2000 Pennington Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Ewing, NJ 08628-0718, USA 9Deptamento de Ingenier´ıa Mec´anica y Minera, Universidad de Ja´en, Campus Las Lagunillas s/n Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' A3 Ja´en,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 23071,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Spain 10CAS Key Laboratory for Researches in Galaxies and Cosmology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' University of Science and Technology of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Hefei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Anhui 230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' China 11School of Astronomy and Space Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' University of Science and Technology of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Hefei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Anhui 230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' China 12Deutsches Elektronen-Synchrotron,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Platanenallee 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' D-15738 Zeuthen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Germany Submitted to ApJS ABSTRACT We present a long-term and intraday variability study on optical multiwaveband (UBVRI) data from the blazar AO 0235+164 collected by various telescopes for ∼44 years (1975–2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The blazar was found to be significantly variable over the years in all wavebands with a variation of about six magnitudes between its low and active states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The variations in the different wavebands are highly correlated without any time-lag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We did not observe any significant trend in color variation with time, but we observed a bluer-when-brighter trend between the B − I color index and the R-magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Optical BVR-band spectral energy distributions always show a convex shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Significant intraday variability was frequently seen in the quasi-simultaneous observations of AO 0235+164 made on 22 nights in R and V -bands by the CASLEO and CAHA telescopes during 1999–2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We also estimated the central supermassive black-hole mass of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9 × 107M⊙ by analyzing the broad Mg II emission line in AO 0235+164’s spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We briefly explore the probable physical scenarios responsible for the observed variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Keywords: galaxies: active – BL Lacertae objects: general – quasars: individual – BL Lacertae objects: individual: AO 0235+164 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' INTRODUCTION Blazars belong to the radio-loud (RL) class of active galactic nuclei (AGNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' This extremely variable class is the union of BL Lacertae objects (BL Lacs) and flat spectrum radio quasars (FSRQs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Blazars host a Corresponding author: Abhradeep Roy abhradeep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1996@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='com, abhradeep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='roy@tifr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='in large-scale relativistic jet of plasma pointing very close to the observer’s line of sight (Urry & Padovani 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The jet is launched from the very near vicinity of the supermassive black hole (SMBH) of mass 106 – 1010 M⊙ at the center of the AGN (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Woo & Urry 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Blazars are characterized by highly variable emission throughout the whole electromagnetic (EM) spectrum, from radio to γ-rays, and their spectral energy distri- butions (SEDs) are characterized by two broad humps (Fossati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Blazars display high and vari- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='01944v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='HE] 5 Jan 2023 ID2 Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' time (JD) 12 13 14 15 16 17 18 19 I mag WEBT-GASP CASLEO-CAHA 12 14 16 18 20 R mag WEBT-GASP Hagen-Thorn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2008 SMARTS Steward Takalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1998 CASLEO-CAHA 14 15 16 17 18 19 20 V mag WEBT-GASP CASLEO-CAHA SMARTS Steward 14 15 16 17 18 19 20 21 B mag WEBT-GASP CASLEO-CAHA SMARTS 2444000 2446000 2448000 2450000 2452000 2454000 2456000 2458000 Time (JD) 16 17 18 19 20 21 U mag WEBT-GASP 1980 1990 2000 2010 2020 Time (Year) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Long-term multiwavelength optical (U, B, V , R, I) lightcurves of AO 0235+164 observed from multiple ground-based telescopes between JD 2442689 (1975 October 3) and JD 2458835 (2019 December 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' AO 0235+164 optical variability 3 able polarization from radio to optical bands, and emit predominately non-thermal emission in the entire EM spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The low-energy hump is ascribed to syn- chrotron radiation from relativistic leptons, whereas the high-energy hump arises from inverse Compton (IC) processes and sometimes from hadronic processes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Marscher 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M¨ucke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Romero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2017, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Blazars display flux variability on diverse timescales ranging from a few minutes to several years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Blazar variability has often been divided into three categories, depending on the cadence of the observations: (i) mi- crovariability (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1989), or intraday variability (IDV) (Wagner & Witzel 1995), or intra-night variabil- ity (INV) (Sagar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2004), focusing on the variability over a day or less;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (ii) short-term variability (STV), focusing on variability over days to weeks, (iii) and long-term variability (LTV), focusing on timescales of months to years (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The BL Lac object AO 0235+164 is at redshift z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='94 (Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Optical spectroscopic and photometric observations of the object have discovered two foreground-absorbing systems at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='524 and z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='851 (Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Nilsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Raiteri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The flux of the source can be both ab- sorbed and contaminated by these foreground systems, and the stars in them may act as gravitational micro- lenses that could contribute to the observed variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Abraham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (1993) did deep CFHT imaging of AO 0235+164 and reported that the source is weakly am- plified by macrolensing / microlensing by stars in the foreground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' AO 0235+164 has been extensively observed in the past from radio to γ-ray bands either in individual EM bands or quasi-simultaneously in multiple EM bands and has shown variations in all those bands on diverse timescales (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Madejski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Rabbette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Takalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Qian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Webb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Romero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Raiteri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2006, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Hagen-Thorn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Agudo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Kutkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Wang & Jiang 2020, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' It is one of the blazars which has displayed very high and variable op- tical/NIR polarization up to ∼45 percent (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Impey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Stickel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Fan & Lin 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Cellone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Ikejiri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Itoh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2016, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' In the Hamburg quasar monitoring program (HQM) this source was observed in the optical R band during 1988–1993, during which a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='36±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='25 magnitude variation was detected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' a particularly strong brightening in the source of ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6 magnitude was re- ported during February 20–22, 1989 (Schramm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' In six nights of optical B and V bands obser- vations during 21–27 September 1992, the blazar was found in an unusually bright state and IDV was de- tected in both B and V bands (Rabbette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' On another occasion, 6 nights of quasi-simultaneous V and R band observations in November 1999, revealed IDV with an amplitude of ∼100 percent over timescales of a day, while 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 magnitude changes were reported in both bands on a single night (Romero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' In multicolor optical/NIR photometric (BVRIJHK) and R-band optical polarimetric observations of AO 0235+164 during its 2006 December outburst, variabil- ity on IDV timescales was detected, with increasing minimum timescale of variability from optical to NIR wavelengths;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' such variations were even detected in the optical polarization (Hagen-Thorn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' In three nights of optical observations of the blazar in January – March 2007, IDV and STV were detected (Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' In quasi-simultaneous optical (V and R bands) and radio (22 GHz) observations of AO 0235+164 during 1993–1996, the variability in optical bands showed am- plitudes up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 magnitudes on STV timescales;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' al- though the radio variability is less dramatic, in general, it followed the optical behavior (Takalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' For the 1997 AO 0235+164 outburst, quasi-simultaneous multi-wavelength (MW) (radio, optical, NIR, and X- ray) observations were carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' It was found that the source varied nearly simultaneously over 6 decades in frequency during the outburst and this result was explained in terms of a microlensing event (Webb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' An analysis of this source’s variability over ∼25 years led to the suggestion of a ∼5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7 years quasi-periodicity of the main radio and optical flares (Raiteri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' however, the putative next outburst, predicted to peak around February–March 2004, did not occur, and a new analysis of the optical light curves on a longer time span revealed a characteristic variability timescale of ∼8 years, which was also present in the radio data (Raiteri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Recently, optical R band photo- metric data taken during 1982–2019 showed 5 cycles of double-peaked periodicity of ∼8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='13 years with a sec- ondary peak following the primary one by ∼(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0) years (Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' In another MW campaign from radio to UV bands in 2006–2007, a huge NIR-optical- UV outburst with brightness increase of ∼5 magnitudes 4 Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' during February 19 – 21, 2007 was detected (Raiteri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' During a major outburst seen in 2009, changes in radio, optical, X-ray, and γ-ray bands were found to be strongly associated (Agudo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' In another simultaneous MW observing campaign of this blazar between 2008 September and 2009 February, γ-ray activity was found to be well correlated with a se- ries of NIR/optical flares, accompanied by an increase in the optical degree of polarization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' the X-ray light curve showed a different 20-day high state of an unusually soft spectrum which did not match the extrapolation of the optical/UV synchrotron spectrum (Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' AO 0235+164 is one of the sources that often used to be called OVV (optically violently variable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' There are several such objects, like 3C 279, 3C 454.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3, 4C 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='45, CTA 102, BL Lacertae, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Long-term achro- maticity and zero lags have widely been found for these sources (Bonning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Raiteri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' AO 0235+164 is peculiar because it is commonly considered a BL Lac, one of the furthest known, but it shares properties with FSRQs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' It is also a complex source because its light is contaminated by the southern AGN, ELISA, and absorbed by an intervening galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' This paper has undertaken a detailed analysis of the source’s optical brightness and spectral variability over a very long time span (∼5 decades) as well as an investiga- tion of its central engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Our aim is to shed light on the long and short-term behavior of an emblematic BL Lac object through a detailed analysis of what is likely the most massive data set ever assembled for an object of this kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' In section 2, we provide descriptions of the observations of AO 0235+164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The section 3 gives our data analy- sis methods and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We present a discussion and conclusions in section 4 and section 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' OBSERVATIONS Most of the optical UBV RI observations of AO 0235+164 we have employed in this work are taken from The Whole Earth Blazar Telescope1 (WEBT) (Villata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Raiteri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2017) which is an in- ternational collaboration of optical, near-infrared, and radio observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' WEBT has organized several monitor- ing campaigns on the blazar AO 0235+164, with the participation of many tens of observers and telescopes all around the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Later, this source was studied 1 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='oato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='inaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='it/blazars/webt by the WEBT and by its GLAST-AGILE Support Pro- gram (GASP) (Villata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2008, 2009), which was started in 2007 to record quasi-simultaneous data of various blazars observed by the AGILE and Fermi (for- merly GLAST) satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' WEBT/GASP data on AO 0235+164 were published in Raiteri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (2001, 2005, 2006, 2008) and Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Raiteri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (2005) prescribed ways to remove the contribution of the southern galaxy ELISA from the observed optical flux densities and estimated the amount of absorption towards the source in excess of that from our Galaxy in X-ray, ultraviolet, optical, and near-infrared bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The WEBT and GASP data were calibrated following a common prescription, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', with the same photome- try for the same reference stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' For calibration of the AO 0235+164 observations, the adopted photometric sequence includes stars 1, 2, and 3 from Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' To build a reliable lightcurve for further anal- ysis, clear outliers were removed and minor systematic offsets between various datasets were corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' AO 0235+164 was also observed with the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2 m tele- scope of Calar Alto Astronomical Observatory (CAHA, Spain) in November – December 2005, using the CAFOS instrument in imaging polarimetry mode, and photo- metric data were obtained by adding up the ordinary and extraordinary fluxes from each individual image (Cellone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Photometric data were also obtained with the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='15 m telescope at Complejo As- tron´omico El Leoncito (CASLEO, Argentina) along several runs in November 1999, December 2000, August 2004, and January 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Results from these data were published in Romero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (1999, 2000, 2002) and in two papers by the WEBT collaboration focused on this blazar (Raiteri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2005, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Data from a more recent (December 2019) observing run with the same telescope were used in Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Magnitude calibration to the standard system was done using our own photometry of Landolt’s (2009) fields as well as standard stars in the field of AO 0235+164 (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Gonz´alez-P´erez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We also collected the publicly available optical R and V -band data of AO 0235+164, taken at Steward Ob- servatory2, University of Arizona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' These measurements employed the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3 m Bok and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='54 m Kuiper telescopes between 4 October 2008 and 12 February 2018, using the SPOL CCD Imaging/Spectropolarimeter attached 2 http://james.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='arizona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='edu/∼psmith/Fermi/DATA/Rphotdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' html AO 0235+164 optical variability 5 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Result of flux variability on optical UBVRI long-term lightcurves of AO 0235+164 Optical Total χ2 red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' χ2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='999,red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Status Variability filter Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' amplitude (%) U 109 904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='47 V 548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8 B 894 3246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='15 V 590.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9 V 1403 5968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='12 V 589.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0 R 5675 8715.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='06 V 718.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8 I 1173 3555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='13 V 567.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 Note—In the fourth column ’V/NV’ represents variable/non- variable status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' to those two telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Details about the instru- ment, observation, and data analysis are given in Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' In addition, we included the optical- BV R data from the Small and Moderate Aperture Research Telescope System (SMARTS) public archive3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The SMARTS consortium is part of the Cerro Tololo Inter-American Observatory (CTIO), Chile, and has been observing Fermi-Large Area Telescope (LAT)- monitored blazars in the optical B, V , R and NIR J and K bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Details about the SMARTS instruments, observations, and data analysis procedures are given in Bonning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' These standard magnitudes observed by CASLEO, CAHA, SMARTS, and the Stew- ard observatory were further corrected for the southern galaxy ELISA following Raiteri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We also added other R-band optical photometric data from the literature (Takalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Hagen-Thorn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' DATA ANALYSIS METHODS AND RESULTS We combined all the optical U, B, V , R, I band data to plot the long term (1974–2020) MW lightcurves of blazar AO 0235+164 (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We removed the ob- servations with errors of more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1 magnitudes and studied long-term and intraday variability, color varia- tion, spectral properties, and inter-band correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Flux variability studies We use different tools on the observed optical magni- tudes to quantify the variability timescales and the cor- responding significance in multiple optical wavebands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The χ2test 3 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='yale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='edu/smarts/glast/home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='php# For a time series of flux density observations, the χ2 is defined as, χ2 = N � i=1 (Mi − ¯ M)2 ε2 i (1) where Mi is the magnitude obtained at the ith observa- tion, εi is the corresponding error in measurement, and ¯ M is the average magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' If the obtained χ2 value is higher than the critical χ2 value at 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9 per cent sig- nificance level, we consider the source as variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The critical value (χ2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='999,d) depends on the degrees of free- dom (d) of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The reduced χ2 values listed in Table 1 indicate that the source exhibits significant flux variations in all the optical wavebands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Variability amplitude According to the relation given by Heidt & Wagner (1996), we estimated the variability amplitudes (VM) in percentage for the lightcurves in different wavelengths using the following formula, VM = 100 × � (Mmax − Mmin)2 − 2 ¯ε2 (%) (2) where Mmax and Mmin are the maximum and minimum observed magnitude in a lightcurve, respectively, while ¯ε is the average error in magnitude measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We list the calculated variability of amplitudes in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Correlation study To study the inter-band correlations, we first gener- ated 15-minute binned optical UBVRI lightcurves, and plotted the average U, B, V , and I-magnitudes against the average R-magnitudes for the time bins when the source was observed at both the wavebands (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The magnitude-vs-magnitude plots show very good linear correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' To take the uncertainty of magni- tude measurements into account, we simulated 10000 datasets assuming that each magnitude measurement is Gaussian distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Then we calculated the mean and standard deviation of the Pearson correlation co- efficients of all simulated datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We obtained high correlations (> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9) with small uncertainties (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='003) between all wavebands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Moreover, to find any time lag between the correlated optical lightcurves we computed the discrete correlation function (DCF) from the unbinned multiwavelength light curves, as the light curves consist of discrete data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Following the method of Edelson & Krolik (1988), we computed the unbinned DCF (UDCF) be- 6 Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 15 16 17 18 R magnitude 17 18 19 20 U magnitude U-mag vs R-mag Pearson coeff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='96±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='93e-03 fit: Umag = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='92*Rmag+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='24 14 15 16 17 18 19 R magnitude 15 16 17 18 19 20 V magnitude V-mag vs R-mag Pearson coeff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='99±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='65e-04 fit: Vmag = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='00*Rmag+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='79 14 15 16 17 18 19 R magnitude 16 17 18 19 20 21 B magnitude B-mag vs R-mag Pearson coeff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='99±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='30e-04 fit: Bmag = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='01*Rmag+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='65 14 15 16 17 18 19 R magnitude 13 14 15 16 17 18 19 I magnitude I-mag vs R-mag Pearson coeff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='99±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='58e-04 fit: Imag = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='98*Rmag-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='64 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 15-minute averaged UBV I magnitudes versus R-magnitude plots for correlation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' U, B, V , and I-band observations show high linear correlation with R-band data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' All the plots are fitted with straight lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' tween the ith data point in one waveband (a) and the jth data point in another (b) as UDCFij = (ai − ¯a)(bj − ¯b) σaσb , (3) where ¯a and ¯b are the mean of the observed magnitudes, and σa and σb are the standard deviations of the cor- responding datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Next, we calculated the discrete correlation function (DCF) at a certain time lag τ by averaging the UDCFijs whose corresponding time lags ∆tij = ta i − tb j lie within the range [τ − ∆τ 2 , τ + ∆τ 2 ] (∆τ is the time lag bin width), such that, DCF(τ) = 1 n � UDCFij(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (4) Following the suggestion of White & Peterson (1994), we computed the mean magnitudes (¯a and ¯b) and the AO 0235+164 optical variability 7 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0 Time lag (days) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3 DCF U vs R 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0 Time lag (days) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='00 DCF B vs R 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0 Time lag (days) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='00 DCF V vs R 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0 Time lag (days) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='00 DCF I vs R Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Results of discrete cross-correlation analysis of U, B, V , and I-band with respect to R-band in the full time range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' standard deviations (σa and σb) in Equation 3 using only those data points who fall within a given time lag bin, as the mean and standard deviation keep on changing for a time series originated from a stochastic process such as blazar emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The error in the DCF(τ) computation in each bin is calculated as σDCF(τ) = 1 M − 1 � � � � M � k=1 (UDCFk − DCF(τ))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (5) Figure 3 shows the DCFs of UBV I bands with respect to the R-band observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' In all cases, the DCFs peak at zero time lag, except the U-band vs R-band DCF due to poor data sampling in the U-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' This explains the strong linearity in Figure 2 and implies that the emission at all optical wavebands are coming from the same region in the jet and are produced from the same radiation mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Color variation with time in optical UBVRI long- term lightcurves of AO 0235+164 CI m c ρ p U − B −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='52E-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='74E+01 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='06E-01 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='28E-02 B − V 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='58E-06 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='52E+01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='42E-01 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='79E-03 V − R −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='34E-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='38E+01 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='19E-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='39E-02 R − I 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='83E-05 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='40E+01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='85E-01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='74E-08 U − I 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='63E-05 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='35E+02 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='03E-01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='88E-03 B − I 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='16E-05 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='92E+01 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='50E-01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='41E-11 Note—In the column headings: CI: color indices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' m = slope;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' c = intercept;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' ρ = Pearson coefficient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' p = null hypothesis probability for Figure 4a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Color Variations The term ‘color’ denotes the magnitude difference be- tween two quasi-simultaneous observations at two dif- 8 Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1 0 1 U-B 0 1 2 B-V 0 1 2 V-R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 Color R-I 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 U-I 2444000 2448000 2452000 2456000 Time (JD) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0 B-I (a) 1 0 1 U-B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6 B-V 0 1 2 V-R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 Color R-I 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 U-I 14 15 16 17 18 19 20 R magnitude 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0 B-I (b) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (a) Color variation with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (b) Color variation with optical R magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The red line in each panel represents the straight line fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Fit parameters are given in Table 2 and Table 3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Color variation with R-band magnitude in optical UBVRI long-term lightcurves of AO 0235+164 CI m c ρ p U − B −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='36E-01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='37E+00 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='37E-01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='35E-05 B − V 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='62E-02 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='04E-01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='41E-01 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='41E-03 V − R −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='54E-03 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='98E-01 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='58E-02 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='92E-01 R − I 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='62E-02 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='00E-01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='37E-01 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='59E-03 U − I −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='47E-02 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='85E+00 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='07E-01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='30E-01 B − I 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='23E-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='66E+00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='66E-01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='69E-07 Note—In the column headings: CI: color indices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' m = slope;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' c = intercept;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' ρ = Pearson coefficient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' p = null hypothesis probability for Figure 4b ferent wavebands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We plotted the variation of optical colors (U − B, B − V , V − R, R − I, and B − I) with time and R-magnitude in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We listed the re- sults of a straight line (Y = mX + c) fitting to all these plots in Table 2 and Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The linear fits of the color versus time plots do not show any trend, except for the rather sparsely sampled (B − I) color, which has a high slope (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='16×10−5) in Figure 4a, along with the highest Pearson correlation coefficient (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='45), and the lowest null hypothesis probability (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='41×10−11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Among the color versus magnitude relations, the strongest relationship is between (B − I) and R (Figure 4b), having a positive slope (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='23×10−2) with the highest Pearson coefficient (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='37) and the lowest p-value (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='69×10−7) (Table 3), in- dicates a bluer-when-brighter (BWB) trend when the widest range of the available colors is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Spectral Variations and SEDs We plotted the optical (BVR) spectral energy distri- butions for the nights where observations were taken at all of these three filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Following the prescription of Raiteri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (2005), we took into account the total absorption by the Milky Way galaxy and the foreground AO 0235+164 optical variability 9 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' An example frame of the AO 0235+164 optical SED animation that is available in the HTML version of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The duration of the animation is 1 minute and it contains a total of 360 one-day averaged optical SEDs, having 6 SEDs per frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The observation dates of the SEDs are given in the plot legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Spetral index variation with R-band magnitude and time in optical UBVRI long-term lightcurves of AO 0235+164 Dependency m c ρ p αV R vs R −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='01E-02 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='30E+00 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='58E-02 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='92E-01 αV R vs JD −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='03E-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='74E+01 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='19E-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='39E-02 Note—In the column headings: m = slope;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' c = intercept;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' ρ = Pearson coefficient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' p = null hypothesis probability for Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' absorber at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='524, and subtracted the extinction magnitudes (AU = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='519, AB = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='904, AV = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='473, AR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='260, AI = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='902) from the calibrated magni- tudes of respective wavebands and then converted them into extinction-corrected flux densities, Fν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The accom- panying video contains one-day averaged optical SEDs for those 360 nights (An example frame is shown in Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Figure 6 shows a few examples of SEDs of low, moderate, and high flux states, plotted in (νFν – ν) format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Mostly, the SEDs have a declining shape following a power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' However, there are evidences of spectral hardening on several nights (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', JD 2445337, JD 2445721, JD 2448889, JD 2452901, JD 2453230).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' From the one-day binned multiwavelength lightcurves we calculated the spectral indices (αV R) for all the days when the source was observed in both V and R bands, AO 0235+164 Optical SEDs 10-10 10-11 JD 2448265 JD 2448266 JD 2448268 JD 2448269 10-12 JD 2448889 JD 2449601 5 × 1014 6 × 1014 7 × 1014 V (HZ)10 Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 5 × 1014 6 × 1014 7 × 1014 8 × 1014 (HZ) 10 12 10 11 10 10 F (erg cm 2 s 1) JD 2449690 JD 2452169 JD 2445343 JD 2451896 JD 2457045 JD 2450811 JD 2454733 JD 2446763 JD 2453230 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Examples of AO 0235+164 optical intraday SEDs during three different states of brightness: (i) the green lines represent SED during quiescent states (νFν (erg cm−2 s−1) < 10−12), (ii) the blue lines show SED during moderately bright states (10−12 < νFν (erg cm−2 s−1) < 3×10−11), (iii) the red lines show SED during outbursts (νFν (erg cm−2 s−1) > 5×10−11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The black lines are examples of SED with spectral hardening on JD 2446763 and JD 2453230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' using the formula given by Wierzcholska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (2015) on extinction corrected magnitudes, as αV R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4(V − R) log(νV /νR) , (6) where νV and νR respectively represent the effective fre- quencies of V and R band filters (Bessell 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We plotted the variation of spectral indices with time and R-band magnitude (Figure 7) and listed the results of linear fits, Pearson coefficient, and null hypothesis prob- ability in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We do not find any significant long- term variation of the spectral index with time, nor is there a correlation with R-magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Intraday Variability We applied four frequently used statistical tests for IDV: scaled C-criterion, scaled F-test, the power-enhanced F- test, and the nested analysis of variance (ANOVA) test (de Diego 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' de Diego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Zibecchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2017, 2020) to detect statistically significant intraday flux variability in AO 0235+164 lightcurves observed by CASLEO and CAHA telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' These tests mainly compare the variations in blazar magnitudes with the variations in magnitudes of one or more stars within the field-of-view of the blazar and have different advan- tages and disadvantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We collected data from mul- tiple field stars along with the blazar data (Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We applied the first three methods on the intraday dif- ferential lightcurves of AO 0235+164 where at least 10 observations were recorded per night with at least one optical filter between 1999 November 2 to 2019 Decem- ber 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We employed the nested ANOVA test only on lightcurves having at least 20 observations per night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Equivalence between internal field star numbering in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='the CASLEO/CAHA data used in the IDV analyses and field- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='star numbering in other standard star charts during different ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='observation seasons ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='Season ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='CASLEO/CAHA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='Heidelberga ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='GKM2001b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1999–2001 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='(CASLEO) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='C1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2004–2005 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='(CASLEO) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='C1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2005 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='(CAHA) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='C1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2018–2019 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='(CASLEO) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='C1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='Note—a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='lsw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='uni-heidelberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='de/projects/ extragalactic/charts/0235+164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='html b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Gonz´alez-P´erez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (2001) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Scaled C-criterion Differential photometry, where the blazar magnitudes are compared to one or more stars in the same field of view, is the usual technique for obtaining blazar lightcurves free from the effects of any non-astrophysical fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The simplest differential photometry in- volves a single comparison star, while a second star, whose magnitudes are measured against the same com- parison star, is used for a stability check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We denote B, S1, and S2 as the blazar, comparison, and control star, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The variability test requires two differen- tial lightcurves (DLC): (blazar–comparison star) and (control star–comparison star).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The latter is believed AO 0235+164 optical variability 11 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Result of scaled C-criterion and F-test for IDV on AO 0235+164 differential lightcurves from CASLEO and CAHA Date JD Band No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' of S1, S2 Γ CΓ FΓ F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='005 c Status Final obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Status 1999 Nov 2 2451485 V 23 2,3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8886 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3640 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1405 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1246 V V 2,6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0867 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9184 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8856 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1912 V 2,10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6876 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1627 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6298 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1246 V 2,11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7431 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4002 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5650 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1246 V 1999 Nov 3 2451486 V 22 2,3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0707 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6976 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4624 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1347 V V 2,11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8841 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0726 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8768 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1347 V 1999 Nov 4 2451487 R 30 2,3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0059 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4058 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6582 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6737 V V 2,11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6639 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8857 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7278 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6737 V V 30 2,3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9994 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9281 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7104 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6737 V V 2,11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8286 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6683 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4770 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6737 V 1999 Nov 5 2451488 R 23 2,3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4994 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5631 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4433 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1246 NV NV 2,11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9852 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9303 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7260 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1246 NV V 22 2,3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4403 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0342 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2064 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1347 V V 1999 Nov 6 2451489 R 30 2,3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8471 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5775 308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9682 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6737 V V 2,6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9769 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3281 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9824 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6737 V 2,7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3573 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9373 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7501 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7048 V 2,8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3805 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8381 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7876 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7048 V 2,10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6936 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8657 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1376 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6737 V 2,11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5616 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4338 238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2019 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6737 V V 29 2,3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8485 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1892 330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8486 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7233 V V 2,6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0013 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7527 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1254 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7233 V 2,7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3527 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5480 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4516 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7397 V 2,8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4133 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4172 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0214 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7397 V 2,10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5626 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6674 312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1376 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7233 V 2,11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7018 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9948 323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8145 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7233 V 1999 Nov 7 2451490 R 11 2,3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9562 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5930 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9095 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8479 V PV 2,4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9798 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2801 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1990 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8479 NV 2,6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1143 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3903 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2751 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8479 V 2,10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9703 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7073 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9148 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8479 NV 2,11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6197 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9496 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7003 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8479 V V 12 2,3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9382 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9304 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5871 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3191 V PV 2,4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7807 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9342 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7410 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3191 NV 2,6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1169 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8931 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3701 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3191 V 2,10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7653 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1046 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4292 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3191 NV 2,11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7772 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3359 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7997 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3191 V Note—S1 and S2 are the comparison and control star numbers, respectively, used for the IDV tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Star numbers follow the star maps shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 12 Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 13 14 15 16 17 18 R magintude 2 0 2 4 6 8 10 VR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='02*R+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='30 (a) 2445000 2447500 2450000 2452500 2455000 2457500 Time (JD) 2 0 2 4 6 8 10 VR 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='03e-05*Time+7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='74e+01 (b) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (a) Variation of spectral index (αV R) with R-band magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (b) Variation of αV R with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The red line at each panel represents the linear fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Result of scaled C-test and F-test for IDV on AO 0235+164 differential lightcurves from CASLEO and CAHA (continued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=') Date JD Band No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' of S1, S2 Γ CΓ FΓ F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='005 c Status Final obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' status 2000 Dec 21 2451900 R 10 2,3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9446 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3638 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5876 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5402 NV PV 2,6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0793 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9877 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8767 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5402 V 2,7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5020 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0187 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0753 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5402 NV 2,8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5289 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9985 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9939 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5402 NV 2,9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8790 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5120 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3100 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5402 NV 2,11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6246 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4168 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0085 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5402 V V 10 2,3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9509 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4671 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0208 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5402 V PV 2,6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1202 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4789 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1449 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5402 NV 2,7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5357 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8729 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5079 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5402 NV 2,8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6064 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0031 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0124 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5402 NV 2,9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0966 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7299 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9120 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5402 V 2,11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7842 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5920 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5343 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5402 NV 2000 Dec 23 2451902 R 10 2,3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8588 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4475 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7803 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5402 V V 2,6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9890 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1629 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6559 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5402 V 2,7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3855 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5919 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9020 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5402 V 2,8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4091 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8222 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9646 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5402 V 2,9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8000 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6739 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8451 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5402 V 2,11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5664 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3690 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8267 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5402 V 2,13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7083 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0181 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1089 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5402 V V 11 2,3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8509 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5241 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5634 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8479 V PV 2,6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0031 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4277 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4602 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8479 V 2,7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3714 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0139 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1395 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8479 V 2,8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4341 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2879 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9619 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8479 V 2,9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9797 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4805 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1919 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8479 NV 2,11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7013 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1765 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7965 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8479 V 2,13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5668 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3770 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1586 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8479 V Note—S1 and S2 are the comparison and control star numbers respectively used for the IDV tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Star numbers follow the star maps shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' AO 0235+164 optical variability 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='80 JD (+2451485) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7 Differential magnitude Date: 1999 Nov 02 [Status: Variable] V band Blazar-S1 (S2-S1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='70 JD (+2453680) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='650 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='675 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='725 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='775 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='825 Differential magnitude Date: 2005 Nov 05 [Status: Variable] R band Blazar-S1 (S2-S1)+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='68 JD (+2451900) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='64 Differential magnitude Date: 2000 Dec 21 [Status: Probably Variable] V band Blazar-S1 (S2-S1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='58 JD (+2453711) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='42 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='44 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='48 Differential magnitude Date: 2005 Dec 06 [Status: Probably Variable] R band Blazar-S1 (S2-S1)+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='80 JD (+2452225) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='650 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='675 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='700 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='725 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='750 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='775 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='800 Differential magnitude Date: 2001 Nov 11 [Status: Non Variable] V band Blazar-S1 (S2-S1)+1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='66 JD (+2458835) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='26 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='28 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='34 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='36 Differential magnitude Date: 2019 Dec 17 [Status: Non Variable] R band Blazar-S1 (S2-S1)+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='55 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Some intraday lightcurves of AO 0235+164 on nights when the source showed different states of variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' S1 and S2 represent the comparison and control star respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' In some panels, the differential lightcurve of the control star is shifted to bring it into the same frame of the blazar DLC for better visual comparison of variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 14 Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Result of scaled C-test and F-test for IDV on AO 0235+164 differential lightcurves from CASLEO and CAHA (continued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=') Date JD Band No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' of S1, S2 Γ CΓ FΓ F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='005 c Status Final obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' status 2001 Nov 9 2452223 R 12 2,11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2042 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6476 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5998 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3191 V V V 12 2,3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8778 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5035 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2675 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3191 NV NV 2,4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6191 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1450 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3111 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3191 NV 2,9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2039 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2380 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5326 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4171 NV 2,10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5871 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0056 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0226 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3191 NV 2,11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5366 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7566 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0857 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3191 NV 2001 Nov 10 2452224 R 10 2,3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3728 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0570 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1172 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5402 NV NV 2,9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2429 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1058 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2229 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5402 NV 2,11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5595 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9395 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8826 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5402 NV V 10 2,3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3876 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0788 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1637 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5402 NV NV 2,9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7847 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3860 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9209 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5402 NV 2,11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9713 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8168 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5402 NV 2001 Nov 11 2452225 R 14 2,3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0291 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4125 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9951 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5724 NV NV 2,9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5447 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2505 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5638 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6425 NV 2,11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3171 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6860 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8427 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5724 NV V 14 2,3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0291 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4125 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9951 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5724 NV NV 2,9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5447 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2505 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5638 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6425 NV 2,11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3171 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6860 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8427 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5724 NV 2001 Nov 12 2452226 R 12 2,3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8479 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5819 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5025 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3191 NV PV 2,11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2074 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0203 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1222 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3191 V V 12 2,3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8704 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9230 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6980 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3191 NV NV 2,4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5981 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0281 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0571 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3191 NV 2,10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5672 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3374 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4634 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3191 NV 2,11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5330 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5642 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4468 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3191 NV 2001 Nov 13 2452227 R 11 3,4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0213 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1434 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3073 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8479 NV NV V 11 3,4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1840 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6858 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4703 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8479 NV NV Note—S1 and S2 are the comparison and control star numbers respectively used for the IDV tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Star numbers follow the star maps shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' AO 0235+164 optical variability 15 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Result of scaled C-test and F-test for IDV on AO 0235+164 differential lightcurves from CASLEO and CAHA (continued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=') Date JD Band No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' of S1, S2 Γ CΓ FΓ F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='005 c Status Final obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' status 2005 Jan 16 2453387 R 11 2,3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5238 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8074 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4962 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8479 V V 2,4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3465 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7388 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5010 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8479 V 2,6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3316 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7442 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5308 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8479 V 2,7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4051 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4058 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5996 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8479 V 2005 Nov 2 2453677 R 32 2,3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2848 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4237 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2636 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5846 V V 2,4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8959 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5227 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4545 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5846 V 2,5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2571 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3013 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5013 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5846 V 2,6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5453 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9310 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4528 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5846 V 2,7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5844 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9283 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2884 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5846 V 2,8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4738 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0560 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7865 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5846 V 2,9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3415 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5374 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7373 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5846 V 2,10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3397 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0828 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6695 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5846 V 2005 Nov 4 2453679 R 12 2,3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8534 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3059 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5409 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3191 V V 2,4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5599 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7978 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4235 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3191 V 2,5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1421 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0805 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8111 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3191 V 2,6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3029 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3341 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4524 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3191 V 2,7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3534 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6846 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0525 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3191 V 2,8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2839 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3153 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6220 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3191 V 2,9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1914 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0664 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4647 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3191 V 2,10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1875 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4019 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1804 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3191 V 2005 Nov 5 2453680 R 44 2,3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9749 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6766 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9894 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2266 V V 2,4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6398 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3431 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2939 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2266 V 2,5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1942 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0439 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9674 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2266 V 2,6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3721 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7338 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2142 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2266 V 2,7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4059 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2100 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2433 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2266 V 2,8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3427 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3494 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7127 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2266 V 2,9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2399 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1459 147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5239 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2266 V 2,10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2340 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5775 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5744 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2341 V 2005 Nov 6 2453681 R 40 2,3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0022 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8517 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6495 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3212 V V 2,4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6946 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8524 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3645 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3212 V 2,5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2051 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6830 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6620 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3212 V 2,6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4022 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6630 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7211 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3212 V 2,8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3694 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9489 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3890 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3212 V 2,9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2576 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8684 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1751 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3212 V 2,10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2563 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1520 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5433 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3212 V Note—S1 and S2 are the comparison and control star numbers respectively used for the IDV tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Star numbers follow the star maps shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 16 Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Result of scaled C-test and F-test for IDV on AO 0235+164 differential lightcurves from CASLEO and CAHA (continued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=') Date JD Band No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' of S1, S2 Γ CΓ FΓ F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='005 c Status Final obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' status 2005 Nov 8 2453683 R 28 2,3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9329 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4256 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8834 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7940 NV NV 2,4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6336 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3363 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4585 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7770 NV 2,5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1788 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7843 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1836 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7770 NV 2,6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3598 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2163 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9120 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7770 NV 2,7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4059 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4945 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2335 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7770 NV 2,8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3451 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1606 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6682 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9002 NV 2,9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2318 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3895 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9307 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7770 NV 2005 Dec 5 2453710 R 20 2,3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4796 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4053 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9748 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4317 NV NV 2,4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0247 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7240 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5242 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4317 NV 2,5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3133 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9355 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8752 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4317 NV 2,6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6030 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1896 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4151 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4317 NV 2,8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5634 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0332 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0674 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4317 NV 2,9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3979 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0716 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1482 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4317 NV 2,10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3915 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8089 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4317 NV 2005 Dec 6 2453711 R 16 2,3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4092 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1709 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7129 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0698 NV PV 2,4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9785 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7432 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0118 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0698 V 2,5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2848 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1323 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5467 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0698 NV 2,6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5570 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7562 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5967 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0698 V 2,7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6157 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8489 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4186 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0698 NV 2,8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5266 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3717 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8815 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0698 NV 2,8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5266 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3717 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8815 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0698 NV 2,9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3691 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4056 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7869 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0698 NV 2,10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3688 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0671 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2727 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0698 NV 2019 Dec 17 2458835 R 30 9,10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3151 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2773 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6315 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6740 NV NV 9,11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7377 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3155 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='7307 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6740 NV 9,12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0425 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='0698 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1445 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6740 NV Note—S1 and S2 are the comparison and control star numbers respectively used for the IDV tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Star numbers follow the star maps shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' AO 0235+164 optical variability 17 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Result of power enhanced F-test and nested ANOVA test for IDV on AO 0235+164 differential lightcurves from CASLEO and CAHA Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Band No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' of Power enhanced F-test Nested ANOVA test Status Variability doubling date Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' amplitude(%) timescale star DOF(ν1,ν2) Fenh F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='005 c DOF(ν1,ν2) F F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='005 c (days) 1999 Nov 2 V 23 2 (22, 87) 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='132 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='209 (5, 17) 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='924 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='075 V 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='103 1999 Nov 3 V 22 2 (21, 42) 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='529 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='540 (5, 16) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='920 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='212 V 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='145 1999 Nov 4 V 30 2 (29, 58) 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='046 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='216 (7, 22) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='922 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='109 V 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='106 R 30 (29, 58) 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='016 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='216 (7, 22) 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='356 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='109 V 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='083 1999 Nov 5 V 22 2 (21, 21) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='207 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='216 (5, 16) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='426 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='212 NV 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='140 R 23 (22, 44) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='951 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='487 (5, 17) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='426 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='075 V 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='335 1999 Nov 6 V 29 2 (28, 166) 211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='363 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='960 (7, 21) 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='114 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='179 V 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='092 R 30 (29, 170) 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='913 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='941 (7, 22) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='686 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='109 V 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='085 1999 Nov 7 V 12 2 (11, 55) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='392 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='854 – – – PV 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='170 R 11 (10, 50) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='413 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='988 – – – PV 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='244 2000 Dec 21 V 10 2 (9, 54) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='813 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='055 – – – PV 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='275 R 10 (9, 54) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='73 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='055 – – – PV 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='428 2000 Dec 23 V 11 2 (10, 70) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='314 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='846 – – – PV 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='200 R 10 (9, 63) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='542 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='989 – – – PV 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='180 2001 Nov 9 V 12 2 (11, 54) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='345 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='863 – – – NV 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='372 R 12 (11, 22) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='91 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='612 – – – PV 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='441 2001 Nov 10 V 10 2 (9, 27) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='152 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='557 – – – NV 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='227 R 10 (9, 27) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='054 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='557 – – – NV 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='660 2001 Nov 11 V 14 2 (13, 38) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='923 – – – NV 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='364 R 14 (13, 38) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='923 – – – NV 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='364 2001 Nov 12 V 12 2 (11, 44) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='212 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='969 – – – NV 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='539 R 12 2 (11, 22) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='928 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='612 – – – PV 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='856 2001 Nov 13 V 11 3 (10, 10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='470 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='847 – – – NV 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='160 R 11 3 (10, 10) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='307 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='847 – – – NV 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='178 2005 Jan 16 R 11 2 (10, 40) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='842 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='117 – – – PV 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='095 2005 Nov 2 R 32 2 (31, 247) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='709 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='868 (7, 24) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='156 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='991 V 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='189 2005 Nov 4 R 12 2 (11, 88) 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='995 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='689 – – – V 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='166 2005 Nov 5 R 44 2 (43, 343) 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='459 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='713 (10, 33) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='301 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='26 V 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='146 2005 Nov 6 R 40 2 (39, 273) 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='755 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='767 (9, 30) 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='95 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='45 V 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='227 2005 Nov 8 R 28 2 (27, 182) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='371 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='965 (6, 21) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='449 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='393 PV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='365 2005 Dec 5 R 20 2 (19, 133) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='067 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='200 (4, 15) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='394 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='803 PV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='391 2005 Dec 6 R 16 2 (15, 120) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='863 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='373 – – – PV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='746 2019 Dec 17 R 30 9 (29, 87) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='453 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='075 (7, 22) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='341 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='109 NV 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='038 Note—Comparison star numbers follow the star maps shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' to be affected only by instrumental fluctuations as any known or suspected variable star can be discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Jang & Miller (1997) and Romero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (1999) in- troduced a parameter C defined as C = σB−S1/σS2−S1, where σB−S1 and σS2−S1 are the standard deviations in blazar DLC and control star DLC, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The blazar is considered to be variable with 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 per cent confidence level if C is greater than a critical value of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='576.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Howell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (1988) pointed out that it is important to select non-variable stars with magnitudes close to the blazar magnitude as comparison and control stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Otherwise, even if the blazar is non-variable, there will be difference between σB−S1 and σS2−S1 due to dif- 18 Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Spectral fitting of AO 0235+164, where the black line is the original spectrum while the green line is the single power law for the fitted continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The inset shows Mg II line fitting where the blue, green, and red lines are the narrow, broad, and total components, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' ferences in photon statistics and other random-noise terms (sky, read-out noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' To use field stars with dif- ferent magnitude levels, Howell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (1988) suggests calculating a correction factor Γ to scale σS2−S1 to the instrumental level of σB−S1 for proper comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Γ can be estimated using the following formula: Γ2 = �NS2 NB �2 � N 2 S1(NB + P) + N 2 B(NS1 + P) N 2 S2(NS1 + P) + N 2 S1(NS2 + P) � (7) where N is the total (sky-subtracted) counts within the aperture, while the sub-indices B, S1 and S2 correspond to N of the blazar, comparison star and control star, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The factor P contains the common noise- terms, as P = npix(Nsky + N 2 RON), where npix is the number of pixels within the aperture, Nsky is the sky level and NRON is the read-out noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We used the me- dian values of N of the objects and sky for calculating Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Thus, the scaled C parameter (CΓ) is defined as CΓ = C Γ = 1 Γ � σB−S1 σS2−S1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (8) The source is considered variable if CΓ ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='576.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Even though the C parameter is not a proper statistic, it re- mains a useful indicator of stability (de Diego 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' de Diego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Zibecchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2017, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Scaled F-test The standard F-statistics parameter is F = σ2 B−S1/σ2 S2−S1, where σ2 B−S1 and σ2 S2−S1 are the vari- ances in blazar DLC and a control star DLC respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The scaled F-statistics FΓ is given as FΓ = F Γ2 = 1 Γ2 � σ2 B−S1 σ2 S2−S1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The F-statistic assumes that the uncertainties in the observations are normally distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' If n(B−S1) and n(S2−S1) are the sizes of the blazar and control star DLC respectively, the number of degrees of freedom in the numerator and denominator of the F-statistic are ν1 = n(B−S1) − 1 and ν2 = n(S2−S1) − 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We calculated FΓ and considered the blazar to be vari- able with 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 per cent confidence if FΓ was greater than the critical value F α c (ν1, ν2) at α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='005 (Zibecchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2017, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Power-enhanced F-test The power-enhanced F -test (PEF) has been used in various recent blazar IDV studies (Pandey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Pandey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2020, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The power- enhanced F-statistic has the advantage of comparing the blazar variance to the combined variance of multiple field stars and is given as (de Diego 2014) Fenh = s2 blz s2c , (9) where s2 blz is the variance of the DLC of the blazar with respect to a reference star, and s2 c is the combined vari- ance of the comparison stars’ DLCs with respect to the reference star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Thus, s2 c is given as s2 c = 1 ��k j=1 nj � − k k � j=1 nj � i=1 s2 j,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (10) Here, k is the total number of available comparison stars in the DLC, nj is the number of observations of the jth comparison star, and s2 j,i is the scaled square deviation of the ith observation of the jth comparison star given as s2 j,i = Γj(mj,i − ¯ mj)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (11) Here Γj is the scale factor of the jth comparison star DLC computed following Equation 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Using the data of the field stars, we first checked the star–star DLCs to identify any spikes due to instru- mental errors or improper removal of cosmic rays, and removed them iteratively if they were more than 3 standard deviations from the mean magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We considered a “well-behaved” star with low fluctuations and an average magnitude close to the blazar as the reference star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The number of degrees of freedom in the numerator and denominator of the F-statistics are AO 0235+164 optical variability 19 ν1 = nblz − 1 and ν2 = ��k j=1 nj � − k, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We calculated Fenh, and considered the blazar to be variable (V) with 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 percent confidence if Fenh was greater than the critical value Fc(ν1, ν2) at α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Nested ANOVA test In the nested analysis of variance (ANOVA) test, DLCs of the blazar are generated with respect to all the com- parison stars used as reference stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The details of this method are given in de Diego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The nested ANOVA test needs a large number of points in the light curves, strongly limiting its application to densely pop- ulated DLCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We divided the DLCs with at least 20 observations into groups such that each group contains 4 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Equation (4) of de Diego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (2015) considers an ideal set of lightcurves where the total number of observations are divisible by the group size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' In most of the DLCs in this work, the total number of observations was not an integral multiple of the group size of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' So, in those cases, the last group contained less than 4 observations, and we calculated the degrees of freedom accordingly to compute the mean square due to groups (MSG) and mean square due to the nested obser- vations in groups (MSO(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The ANOVA F-statistic is given as, F = MSG/MSO(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' For a significance level of α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='005, if the F -statistic is greater than the critical value (Fc), the blazar is taken as variable (V), other- wise as non-variable (NV) with 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5 per cent confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We have listed the results of the scaled C-criterion and scaled F-test in Table 6 and those of power en- hanced F-test and the nested ANOVA test in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' In the case of scaled C-criterion and F-test, we fixed one particular star as the comparison star for each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The source is declared variable with respect to one comparison-control star pair if both scaled C-statistics and F-statistics cross their respective critical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We declare the final variability status of the blazar as variable/non-variable (V/NV) if it is variable/non- variable against all control stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' If the blazar is variable against some of the control stars, we call it probably variable (PV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We did not carry out the nested ANOVA test in a few datasets containing less than 20 obser- vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' In the case of the power-enhanced F-test in absence of the corresponding nested ANOVA test, we call the blazar probably variable (PV) even if the F- statistic crosses the critical value, as the F-test is more prone to give a false positive result (Zibecchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2017, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' If nested ANOVA is present and both the tests cross the critical values, we call the blazar variable (V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Otherwise, we declare the source non-variable (NV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We list the summary of the IDV tests in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We give a final verdict on the variability status of the source after comparing the results of the combination of the C-test and F-test (C&F) from Table 6 and results of the combination of the power-enhance F-test and nested ANOVA test (P&N) from Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' If the results from both combinations were the same, we kept that result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' If C&F declared “V” and P&N declared “PV” due to the absence of nested ANOVA, we finally consider the source variable (V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We considered variability on 2005 November 8 as “NV” because both C-test and nested ANOVA resulted in non-variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Despite being vari- able in nested ANOVA, we consider the 2005 December 5 lightcurve “NV” as the F-test and PEF-test detected no variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' A few examples of DLCs of AO 0235+164 having different variability characteristics (V/PV/NV) are shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Doubling timescale A flux doubling/halving timescale gives an estimate of the variability timescale (τvar) of a source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We calcu- late the flux doubling/halving timescale (τd) between two consecutive observations and its corresponding sig- nificance (σ) as F(ti+1) = F(ti) ∗ 2(ti+1−ti)/τd σ = |F(ti+1) − F(ti)|/εi, (12) where F(ti) and εi are the flux observed at time ti and the corresponding measurement uncertainty, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We consider the fastest doubling timescale (τ min d ) with a higher significance than 3σ as an estimate for τvar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We obtained τ min d < 1 day for all the nights when the source showed significant IDV both in scaled F-test and nested ANOVA test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' This further strengthens our claims for the frequent presence of IDV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Following Equa- tion 2 we computed the variability amplitudes on the same nights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' All these results are listed in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Duty cycle We calculated the duty cycle (DC) of AO 0235+164 using the definition of Romero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (1999), that was used later by multiple authors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Stalin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The formula for DC for a partic- ular waveband is given as, DC = 100 �n i=1 Ni(1/∆ti) �n i=1(1/∆ti) % (13) where ∆ti = ∆ti,obs/(1+z) (duration of the monitoring session on ith night is ∆ti,obs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Thus, this formula cal- culates the duty cycle weighted by the cosmological red- shift corrected monitoring duration of each night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We set Ni = 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5, and 0 for the nights with variability 20 Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Summary of statistical tests for IDV on AO 0235+164 differential lightcurves from CASLEO and CAHA Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='Band ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='Combined variability status ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='Final ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='date ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='(C & F-test)a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='(PEF & ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='status ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='N-ANOVA)b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1999 Nov 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1999 Nov 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1999 Nov 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1999 Nov 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1999 Nov 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1999 Nov 7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2000 Dec 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2000 Dec 23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2001 Nov 9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2001 Nov 10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2001 Nov 11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2001 Nov 12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2001 Nov 13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2005 Jan 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2005 Nov 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2005 Nov 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2005 Nov 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2005 Nov 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2005 Nov 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2005 Dec 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2005 Dec 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2019 Dec 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='NV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='Note—aTable 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' bTable 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' PEF=power-enhanced F-test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' status “V”, “PV”, and “NV” respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We obtained the duty cycle of AO 0235+164 to be ∼44 percent in V - band, and ∼45 percent in R-band considering the nights where the source was observed for at least 2 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The mass of the central black hole We estimate the mass of the SMBH in AO 0235+164 by using its spectrum observed using the CCD Imag- ing/Spectropolarimeter (SPOL) at the Steward Obser- vatory4 on 2011 January 8 (air mass = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' This spectrum was selected since the blazar was then at its lowest level during the period 2008–2018, and should ensure the best visibility of the emission lines because of the lower continuum contribution from the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The observed wavelength range of the spectrum we used is 4000–7550 ˚A, with a spectral resolution of 4 ˚A, and it is analyzed by following the procedure given in Liao & Gu (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Firstly, it was corrected for Galactic extinction with the reddening map of Schlegel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (1998), and then was shifted to the rest-frame wavelength by using the redshift of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' This spectral coverage meant we could use the Mg II line, which is prominent on the spectrum shown in Figure 9 (focused on the 2400−3100 ˚A range), to es- timate the SMBH mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We modeled the continuum by applying a single power law (fλ ∝ λα) (as Fe II emission is rather weak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' A Gaussian profile was then used to fit the Mg II line, centered at the position of 2800 ˚A, on the continuum-subtracted spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The broad component of Mg II was fitted with a Gaussian with a 1000 km s−1 lower limit, while a Gaussian with upper limit of 1000 km s−1 was applied for the narrow component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' In order to estimate the corresponding errors of full width at half maximum (FWHM) and flux, we generated 100 mock spectra by adding random Gaussian noise to the original spectrum using the flux density errors, and then took the standard deviation of measurements from those mock spectra as the uncer- tainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Here, the flux density errors were the RMS value of the spectrum calculated over the spectral win- dow of (3000−3100) ˚A, after subtracting a second-order polynomial function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Figure 9 shows the resulting fit to the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Our best fitting results indicate that the line width of the broad Mg II component is FWHM = 3151 km s−1, with log-scale luminosity in erg s−1, log(LMgII) = 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The line width and the Mg II line luminosity we find are consistent with the range of values FWHM=3100– 3500 km s−1 and log(LMgII)=42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5–42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8, respectively, which were derived by Raiteri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (2007) from one VLT and four TNG spectra of AO 0235+164 acquired in 2003–2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We use the FWHM and luminosity of the broad Mg II line, not the continuum luminosity, as 4 http://james.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='arizona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='edu/∼psmith/Fermi AO 0235+164 optical variability 21 1015 1016 (HZ) 10 14 10 13 10 12 F (erg cm 2 s 1) Disk thermal U B V R I JD 2452169 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Comparison of the SED of the lowest flux state observed on JD 2452169 and the thermal emission from the accretion disk in the observer’s frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The thermal emis- sion component is calculated using a multi-temperature disk model with the black hole mass log(MBH/M⊙) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='25, and the log-scale disk luminosity in erg s−1, log(Ldisk) = 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='01±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The shaded region indicates the uncertainties in the calculation of the disk thermal component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' we are unable to exclude the jet emission contribution, despite the low state spectrum that we could use for this blazar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The black hole mass is derived from the empirical relation used for Mg II (Kong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2006), which is based on measured broad line region sizes in the reverberation-mapping AGN sample of Peterson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (2004), as MBH M⊙ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9×106 � LMgII 1042 erg s−1 �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='57±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='12 �FWHMMgII 103 km s−1 �2 (14) Thus, the SMBH mass is log(MBH/M⊙) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='25, where the uncertainty is estimated from the measure- ment uncertainties of the FWHM and luminosity of Mg II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Using optical spectroscopy data from the SDSS archive, Paliya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (2021) reported a somewhat higher mass, log(MBH/M⊙) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='34, and an accretion disk luminosity (in erg s−1), of log(Ldisk) = 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Using the method mentioned in Paliya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (2021) with log(LMgII) = 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='8, we obtained a lower disk lumi- nosity (in erg s−1) of log(Ldisk) = 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='20 from the spectrum observed on 2011 January 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' DISCUSSION In this work, we have presented a detailed temporal and spectral study of the highly variable emission from the blazar AO 0235+164 observed at multiple optical wavebands (UBVRI) from October 1975 to December 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The lightcurves have highly uneven data sampling due to gaps in observation seasons and non-uniform ob- servation campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Although U-band data are quite sparsely sampled the BVRI observations have denser sampling when the source was highly active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Multiple long-term studies suggested that AO 0235+164 shows ∼2-year long flaring episodes with multiple sub-flares after intervals of ∼8 years (Raiteri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Figure 1 shows a difference of about six magnitudes between the quiescent and out- burst states in all optical wavebands, corresponding to an energy flux variation of more than two orders of magnitude (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The long-term variability ampli- tudes at all five wavebands are quite similar (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Also, we found a strong correlation with zero time-lag between the UBVI observations and the R-band data (Figure 2 and Figure 3), which implies a common ra- diative process at a single emission zone is responsible for the bulk of the emission at the optical wavebands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Sometimes during the quiescent states of powerful blazars, the disk thermal emission component becomes visible as a big blue bump on top of the synchrotron emission component from the jet in the optical-UV wavebands (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' As the disk emission is bluer than the jet synchrotron emission, an increase in the jet activity during low flux states displays a redder-when-brighter trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The enhanced jet activ- ity is observed when the charged particles inside the jet get accelerated to higher energies, and then radiate faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Thus, the jet synchrotron component tends to get bluer with the increase in flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' If the jet emission completely outshines the disk emission, we expect to see a bluer-when-brighter trend (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Isler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The flux increment can also be attributed to the in- crease in the jet Doppler factor (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Papadakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2007), which blueshifts the spectrum and produces a bluer-when-brighter trend because of the convexity of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Such a trend is seen in the (B − I) vs R magnitude diagram (Figure 4b) and indicates the dom- ination of non-thermal jet emission over the thermal emission component of the accretion disk during both flaring and quiescent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' From the convex shapes of the optical BVR SEDs during states ranging from quiescent to flaring (see the accompanying SED video and Figure 6), we may infer that the effect of the disk thermal emission is not significant in optical wavebands even during the low flux states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' This can be explained in terms of the nature of disk thermal emission given the disk luminosity and the cen- tral black hole mass computed in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The primary, and most precise, black hole mass estimation methods are based on stellar and gas kinematics and reverberation mapping (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Vestergaard 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' These 22 Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' methods need high spatial resolution spectroscopy data from the host galaxy and/or higher ionization emission lines and are not applicable to most BL Lacertae objects (BL Lacs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' But in BL Lacs, if the weak emission lines are present, we can use the empirical methods (Kong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2006) for BH mass estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The most common methods used for BH mass estimation for BL Lacs are the shortest variability timescales and periods of QPOs (Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Since BL Lacs are highly variable objects, any BH mass estimation may be treated as an upper limit, and there are possibilities of detection of a shorter variability timescale or shorter QPO period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We obtained a log-scale BH mass of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='90±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='25 in so- lar mass unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The Steward observatory spectrum we used in our analysis had a narrower Mg II emission line (FWHM=3151 km s−1) than those of Raiteri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (2007) and Paliya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (2021), thus resulting in a lower mass estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We considered a multi-temperature blackbody type accretion disk model, where the temper- ature at any portion of the disk is a function of the disk luminosity and the central black hole mass, to compute the thermal emission component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' In Figure 10 we plot- ted the thermal component along with the optical-UV SED during the lowest activity state of AO 0235+164 observed on JD 2452169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' It is evident that, as the ther- mal emission peaks at far UV frequencies (∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5×1015 Hz) in the observer’s frame of reference, the jet emission always dominates in BVRI wavebands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We do not see any significant trend in the variation of the (V − R) spectral index (αV R) (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The sudden rise of the U-band flux in Figure 10 is an indicator of a prob- able UV-soft X-ray bump as discussed in Raiteri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (2005, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' According to these studies, the source of the bump is either an additional synchrotron com- ponent coming from a separate emission region in the jet or the emission of a continuous inhomogeneous jet is suppressed in near UV region due to a discontinuity in opacity or misalignment of that particular emission region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (2012) mentioned that the whole optical-UV spectrum is produced by a single syn- chrotron emitting zone as the shape of the bump does not change with luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' They attributed the UV spectral hardening to an artifact due to the overestima- tion of extinction by Junkkarinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' For the detection of any statistically significant intraday variability in 33 lightcurves of AO 0235+164 observed at CASLEO/CAHA, we employed different statistical tests widely used in AGN variability studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The re- liability of each of these tests has been disputed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' de Diego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Zibecchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2017), so we here employed a comparative approach that could allow us to circumvent the limitations affecting any individual test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' In the first place, we used the scaled C-criterion and the F-test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The first compares the dispersion of the blazar lightcurve to the dispersion of a field star (con- trol star), while the latter does so with the variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' According to Zibecchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (2017) and Zibecchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (2020), the F-test has a tendency to classify noisy non- variable curves as a variable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', give false positives), while the C-criterion tends to give false negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Even though the C-criterion (Romero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1999) cannot be considered as an actual statistical test, it may still be a useful parameter to detect variability with high signifi- cance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The F-test, on the other hand, does not always work as expected, because it is particularly sensitive to non-Gaussian errors (“red noise”), which are usually an issue when analyzing blazars DLCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We also used the power-enhanced F-test and the nested ANOVA test, which involve multiple field stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' It is ex- pected that the power-enhanced F-test may also suffer from the same drawback of detecting false variability as the (original) F-test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' In the nested ANOVA test, in turn, data grouping may lead to false results if data within a time span larger than the (unknown) variability timescale are grouped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Comparing the results of Table 6 and Table 7, while considering the tendencies of giving false results by the respective tests, we can confirm that the source was significantly variable in 4 out of 13 V - band lightcurves, and 9 out of 20 R-band lightcurves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The source seems to be probably variable in 3 V -band and 4 R-band lightcurves, and non-variable in the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' On 1999 November 5, the combination of C-criterion and F-test indicates non-variability but the combination of power-enhanced F-test and nested ANOVA detects variability in the R-band lightcurve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The results in the V -band lightcurve on that day are exactly the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Similar situations were observed also on 2001 November 9 and 2001 November 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' A visual inspection of the DLCs of these nights reveals that the blazar DLCs were classified as non-variable when either the control star DLC had higher variability (1999 November 5) or the measurement errors of the blazar DLCs were higher due to its low-flux state (2001 November 9 and 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Higher measurement errors lead to a lower chance of signifi- cant variability detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' These strange results may be an example of the drawbacks of the applied methods when trying to recover low-amplitude variations from DLCs affected by non-Gaussian noise (part of the ob- servations on that night were taken at air mass > 2 and under non-photometric conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Otherwise, the combined results of different methods seem to more or less agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Alongside the optical SED patterns, such frequent IDV establishes AO 0235+164 as a low-energy AO 0235+164 optical variability 23 Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Variation of duty cycle with the duration of observation in R-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Observation No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' of Duty duration (hours) nights cycle (%) > 1 20 52 > 2 19 45 > 3 17 50 > 4 14 57 > 5 13 64 > 6 8 77 peaked BL Lac (LBL) object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' High energy peaked BL Lacs (HBL) show significantly less optical intraday vari- ability than the LBLs (Heidt & Wagner 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Romero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The differences in IDV behavior have been attributed to the strength of magnetic fields present in the jet of HBLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' A higher axial magnetic field (B) than a critical value (Bc) may prevent the generation of any bends and Kelvin-Helmhotz instabilities in the jet-base responsible for creating intraday microvariabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' This indicates the presence of a weaker magnetic field than Bc in the jet of AO 0235+164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' The critical magnetic field (Bc) is given in Romero (1995) as Bc = � 4πnemec2(Γ2 − 1)/Γ, (15) where ne is the electron density in the emission region, me is the electron rest mass, and here Γ is the bulk Lorentz factor of the jet flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Considering a typical set of parameters, ne = 429 cm−3 and Γ = 20 (Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2012), we get Bc ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='07 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' From Table 7 and Figure 8, we can say that the vari- ability amplitudes were higher in the 1999 season when the source was in a fainter state (higher magnitude) than its brighter state in the 2005 season.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Marscher (2013) suggested that enhancement of flux can arise from a more uniform flow of particles inside the jet, which in turn decreases the amplitude of microvariability asso- ciated with the turbulence inside the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Equation 9 indicates that the probability of detection of significant variability increases with the duration of observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Similar results for other blazars were found by Gupta & Joshi (2005), Rani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (2010), and Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' From the flux doubling timescales listed in Table 7, we can estimate the upper limit to the size of the emission region (Rmax) using the light travel-time argument given as Rmax = cδtvar 1 + z (16) where z is the cosmological redshift of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='94, tvar is the variability timescale, and δ is the Doppler boost of the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Considering δ = 24 (Hovatta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2009) and tvar to be the shortest flux doubling timescale of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='083 days (when the source was significantly variable), we obtain an emission region size upper limit of ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='6 × 1015 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Assuming a conical jet model where the emission re- gion fills up the entire jet cross-section, we can estimate the probable maximum distance (dmax) of the emission region from the central black hole as, dmax = ΓRmax = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2×1016 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' To explain the observed strong variability, Marchesini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (2016) attempted to apply a swinging jet model that attributes the observed variability to a change in the viewing angle of the emission region with time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' variation in the associated bulk Doppler fac- tor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' They reported a high rate of change in viewing an- gle of about 7−10 arcmin per day, considering a mean viewing angle of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3◦, would be necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' However, they found that this geometric wiggling-jet scenario was disfavored when considering the observed variation in color index with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Several earlier studies on AO 0235+164 associated the observed fast optical variabil- ity with gravitational microlensing by the foreground absorber at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='524.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Webb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (2000) proposed that the 1997 flare resulted due to microlensing because of an observed correlation with zero lag between radio and op- tical lightcurves following Stickel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (1988), but the absence of any correlated flare in the X-ray lightcurve makes this explanation less likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Abraham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (1993) and Raiteri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' (2007) explained that such microlens- ing events can produce small amounts of fast flux ampli- fication but are unlikely to dominate the high variability observed in AO 0235+164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' CONCLUSIONS In this work, we conducted a study of long-term and short-term (intraday) variability in the optical mul- tiwaveband observations of the blazar AO 0235+164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Here we summarize our results and the probable physi- cal scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We observed a variation of about six magnitudes between the quiescent and flaring episodes, or over two orders of magnitude variation in the SEDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' UBVI lightcurves are highly correlated with the R-band lightcurve with zero time lag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 24 Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' A significant bluer-when-brighter trend is observed in the (B − I) color variation with R-magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' All the optical BVR-band SEDs show convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' These observations indicate that the optical emis- sion is dominated by jet radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' AO 0235+164 frequently shows statistically sig- nificant intraday variability in optical wavebands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' This implies that AO 0235+164 is an LBL and probably has a weak magnetic field in the jet en- vironment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' From the analysis of a broad Mg II emission line in a spectrum of AO 0235+164 taken at a low state, we estimate a central black-hole mass of ∼ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='9 × 107M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' ACKNOWLEDGMENTS Data from the Steward Observatory spectropolari- metric monitoring project were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' This pro- gram is supported by Fermi Guest Investigator grants NNX08AW56G, NNX09AU10G, NNX12AO93G, and NNX15AU81G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' This paper has made use of up-to- date SMARTS optical/near-infrared light curves that are available at www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='yale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='edu/smarts/glast/home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' php.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' This work is partly based on data taken and as- sembled by the WEBT collaboration and stored in the WEBT archive at the Osservatorio Astrofisico di Torino INAF (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='oato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='inaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='it/blazars/webt/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' These data are available upon request to the WEBT President Massimo Villata (massimo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='villata@inaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' This work is based on data acquired at Complejo Astron´omico El Leoncito, operated under an agree- ment between the Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas de la Rep´ublica Argentina and the National Universities of La Plata, C´ordoba and San Juan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We thank Anabella Araudo and Ileana Andru- chow for help with the observations made with CASLEO and the data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We thankfully acknowledge the anonymous reviewer for very useful comments which helped us to improve the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' We acknowledge the support of the Department of Atomic Energy, Government of India, under project identification number RTI 4002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' ACG is partially supported by Chinese Academy of Sciences (CAS) President’s International Fellowship Initiative (PIFI) (grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2016VMB073).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' GER acknowl- edges support from grants PIP 0554 (CONICET), PIP 2021-1639 (CONICET), and grant PID2019- 105510GBC31 of the Spanish Ministerio de Ciencia, Innovaci´on y Universidades and through the Center of Excellence Mara de Maeztu 2020-2023 award to the ICCUB (CEX2019-000918-M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' JAC is Mar´ıa Zam- brano researcher fellow funded by the European Union NextGenerationEU- (UJAR02MZ), supported by PIP 0113 (CONICET) and PICT-2017-2865 (ANPCyT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' JAC was also supported by grant PID2019-105510GB- C32/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='13039/501100011033 from the Agencia Es- tatal de Investigaci´on of the Spanish Ministerio de Ciencia, Innovaci´on y Universidades, and by Consejer´ıa de Econom´ıa, Innovaci´on, Ciencia y Empleo of Junta de Andaluc´ıa as research group FQM-322, as well as FEDER funds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Facilities: WEBT, SMARTS, Bok, SO:Kuiper, MMT, CASLEO:JST, CAO:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2m AO 0235+164 optical variability 25 Software: Astropy (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2013), DAOPHOT (Stetson 1987), IRAF (Tody 1986) REFERENCES Abraham, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Crawford, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Merrifield, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Hutchings, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & McHardy, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1993, ApJ, 415, 101, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1086/173147 Ackermann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Ajello, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Ballet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2012, ApJ, 751, 159, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1088/0004-637X/751/2/159 Ackermann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Ajello, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Ballet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2012, ApJ, 751, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1088/0004-637X/751/2/159 Agarwal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Gupta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Bachev, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2016, MNRAS, 455, 680, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1093/mnras/stv2345 Agudo, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Marscher, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Jorstad, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2011, ApJL, 735, L10, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1088/2041-8205/735/1/L10 Astropy Collaboration, Robitaille, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Tollerud, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2013, A&A, 558, A33, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1051/0004-6361/201322068 Bessell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2005, ARA&A, 43, 293, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1146/annurev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='082801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='100251 Bonning, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Urry, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Bailyn, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2012, The Astrophysical Journal, 756, 13, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1088/0004-637X/756/1/13 Cellone, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Romero, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Combi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Mart´ı, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2007, MNRAS, 381, L60, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1745-3933.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='00366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='x Cohen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Smith, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Junkkarinen, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Burbidge, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1987, ApJ, 318, 577, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1086/165393 de Diego, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2014, AJ, 148, 93, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1088/0004-6256/148/5/93 de Diego, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Polednikova, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Bongiovanni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2015, AJ, 150, 44, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1088/0004-6256/150/2/44 Edelson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Krolik, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1988, ApJ, 333, 646, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1086/166773 Fan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Lin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1999, ApJS, 121, 131, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1086/313191 Fan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Tao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Qian, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2006, Publications of the Astronomical Society of Japan, 58, 797, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1093/pasj/58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='797 Fan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Kurtanidze, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2017, ApJ, 837, 45, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3847/1538-4357/aa5def Fossati, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Maraschi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Celotti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Comastri, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Ghisellini, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1998, MNRAS, 299, 433, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1046/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1365-8711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='01828.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='x Gonz´alez-P´erez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Kidger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Mart´ın-Luis, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2001, AJ, 122, 2055, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1086/322129 Guo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Hu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Xu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2015, NewA, 36, 9, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='newast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='011 Gupta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Banerjee, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Ashok, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Joshi, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2004, A&A, 422, 505, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1051/0004-6361:20040306 Gupta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Fan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Bai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Wagner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2008, AJ, 135, 1384, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1088/0004-6256/135/4/1384 Gupta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Joshi, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2005, A&A, 440, 855, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1051/0004-6361:20042370 Gupta, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Pandey, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Singh, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2012, NewA, 17, 8, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='newast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='005 Hagen-Thorn, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Larionov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Jorstad, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2008, ApJ, 672, 40, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1086/523841 Heidt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Wagner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1996, A&A, 305, 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='org/abs/astro-ph/9506032 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1998, A&A, 329, 853.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='org/abs/astro-ph/9709116 Hovatta, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Valtaoja, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Tornikoski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & L¨ahteenm¨aki, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2009, A&A, 494, 527, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1051/0004-6361:200811150 Howell, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Mitchell, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Warnock, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1988, AJ, 95, 247 Ikejiri, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Uemura, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Sasada, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2011, PASJ, 63, 639, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1093/pasj/63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='327 Impey, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Brand, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Tapia, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1982, MNRAS, 198, 1, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1093/mnras/198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1 Isler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Urry, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Coppi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2017, The Astrophysical Journal, 844, 107, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3847/1538-4357/aa79fc Itoh, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Nalewajko, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Fukazawa, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2016, ApJ, 833, 77, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3847/1538-4357/833/1/77 Jang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Miller, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1997, AJ, 114, 565, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1086/118493 Junkkarinen, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Cohen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Beaver, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2004, ApJ, 614, 658, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1086/423777 Kong, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Wu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Wang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Han, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2006, ChJA&A, 6, 396, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1088/1009-9271/6/4/02 Kutkin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Pashchenko, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Lisakov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2018, MNRAS, 475, 4994, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1093/mnras/sty144 Landolt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2009, AJ, 137, 4186, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1088/0004-6256/137/5/4186 Liao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Gu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2020, MNRAS, 491, 92, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1093/mnras/stz2981 Madejski, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Takahashi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Tashiro, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1996, ApJ, 459, 156, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1086/176877 Marchesini, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Andruchow, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Cellone, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2016, A&A, 591, A21, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1051/0004-6361/201527632 26 Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Marscher, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1983, ApJ, 264, 296, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1086/160597 Marscher, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2013, The Astrophysical Journal, 780, 87, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1088/0004-637x/780/1/87 Miller, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Carini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Goodrich, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1989, Nature, 337, 627, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1038/337627a0 M¨ucke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Protheroe, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Engel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Rachen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Stanev, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2003, Astroparticle Physics, 18, 593, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1016/S0927-6505(02)00185-8 Nilsson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Charles, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Pursimo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1996, A&A, 314, 754 Paliya, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Dom´ınguez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Ajello, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Olmo-Garc´ıa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Hartmann, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2021, ApJS, 253, 46, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3847/1538-4365/abe135 Pandey, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Gupta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Wiita, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Tiwari, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2019, ApJ, 871, 192, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3847/1538-4357/aaf974 Pandey, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Gupta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Kurtanidze, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2020, The Astrophysical Journal, 890, 72, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3847/1538-4357/ab698e Papadakis, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Villata, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Raiteri, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2007, A&A, 470, 857, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1051/0004-6361:20077516 Peterson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Ferrarese, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Gilbert, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2004, ApJ, 613, 682, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1086/423269 Qian, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Kraus, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Witzel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Krichbaum, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Zensus, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2000, A&A, 357, 84 Rabbette, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', McBreen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Steel, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Smith, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1996, A&A, 310, 1 Raiteri, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Villata, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Capetti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2007, A&A, 464, 871, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1051/0004-6361:20066599 Raiteri, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Villata, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Aller, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2001, A&A, 377, 396, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1051/0004-6361:20011112 Raiteri, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Villata, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Ibrahimov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2005, A&A, 438, 39, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1051/0004-6361:20042567 Raiteri, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Villata, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Kadler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2006, A&A, 459, 731, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1051/0004-6361:20065744 Raiteri, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Villata, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Larionov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2008, A&A, 480, 339, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1051/0004-6361:20079044 Raiteri, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Villata, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Acosta-Pulido, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2017, Nature, 552, 374, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1038/nature24623 Rani, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Gupta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Strigachev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2010, Monthly Notices of the Royal Astronomical Society, 404, 1992, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1365-2966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='16419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='x Romero, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1995, Ap&SS, 234, 49, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1007/BF00627281 Romero, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Boettcher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Markoff, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Tavecchio, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2017, SSRv, 207, 5, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1007/s11214-016-0328-2 Romero, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Cellone, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Combi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1999, A&AS, 135, 477, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1051/aas:1999184 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2000, A&A, 360, L47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='org/abs/astro-ph/0007407 Romero, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Cellone, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Combi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Andruchow, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2002, A&A, 390, 431, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1051/0004-6361:20020743 Roy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Patel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Sarkar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Chatterjee, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Chitnis, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2021, MNRAS, 504, 1103, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1093/mnras/stab975 Roy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Chitnis, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Gupta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2022, MNRAS, 513, 5238, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1093/mnras/stac1287 Sagar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Stalin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Gopal-Krishna, & Wiita, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2004, MNRAS, 348, 176, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1365-2966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='07339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='x Schlegel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Finkbeiner, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Davis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1998, ApJ, 500, 525, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1086/305772 Schramm, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Borgeest, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Kuehl, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1994, A&AS, 106, 349 Smith, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Balonek, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Heckert, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Elston, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Schmidt, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1985, AJ, 90, 1184, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1086/113824 Smith, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Montiel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Rightley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2009, arXiv e-prints, arXiv:0912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3621.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='org/abs/0912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3621 Stalin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Kawabata, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Uemura, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2009, Monthly Notices of the Royal Astronomical Society, 399, 1357, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1365-2966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='15354.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='x Stetson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1987, Publications of the Astronomical Society of the Pacific, 99, 191, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1086/131977 Stickel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Fried, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Kuehr, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1988, A&A, 198, L13 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1993, A&AS, 98, 393 Takalo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Sillanpaeae, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Valtaoja, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1998, A&AS, 129, 577, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1051/aas:1998205 Tody, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1986, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 627, Instrumentation in astronomy VI, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Crawford, 733, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1117/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='968154 Urry, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Padovani, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1995, PASP, 107, 803, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1086/133630 Vestergaard, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2004, in Astronomical Society of the Pacific Conference Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 311, AGN Physics with the Sloan Digital Sky Survey, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Richards & P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' Hall, 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='org/abs/astro-ph/0401436 Villata, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Raiteri, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Kurtanidze, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2002, A&A, 390, 407, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1051/0004-6361:20020662 Villata, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Raiteri, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Larionov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2008, A&A, 481, L79, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1051/0004-6361:200809552 Villata, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Raiteri, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Gurwell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2009, A&A, 504, L9, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1051/0004-6361/200912732 Wagner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Witzel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1995, ARA&A, 33, 163, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1146/annurev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='aa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='090195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='001115 Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Jiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2020, ApJ, 902, 41, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='3847/1538-4357/abb36c Webb, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Howard, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Ben´ıtez, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2000, AJ, 120, 41, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1086/301432 AO 0235+164 optical variability 27 White, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Peterson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 1994, PASP, 106, 879, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1086/133456 Wierzcholska, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Ostrowski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Stawarz, �L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Wagner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Hauser, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2015, A&A, 573, A69, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1051/0004-6361/201423967 Woo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Urry, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2002, ApJ, 579, 530, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1086/342878 Zhang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Jin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Zhao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Dai, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2021, Research in Astronomy and Astrophysics, 21, 186, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1088/1674-4527/21/8/186 Zibecchi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Andruchow, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Cellone, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', & Carpintero, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2020, MNRAS, 498, 3013, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1093/mnras/staa2544 Zibecchi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Andruchow, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', Cellone, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content=' 2017, MNRAS, 467, 340, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} +page_content='1093/mnras/stx054' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQf_P4t/content/2301.01944v1.pdf'} diff --git a/99E1T4oBgHgl3EQfCgIZ/content/2301.02864v1.pdf b/99E1T4oBgHgl3EQfCgIZ/content/2301.02864v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c3cd3dd8064910e0edfa11dc1bb676d1fb02eded --- /dev/null +++ b/99E1T4oBgHgl3EQfCgIZ/content/2301.02864v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0f33f60c2ae788ddc7a38b0ade1050c8b2d2cbfd72cf43d4a114a041cfce3080 +size 1948945 diff --git a/99E1T4oBgHgl3EQfCgIZ/vector_store/index.pkl b/99E1T4oBgHgl3EQfCgIZ/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..cc5fdb094f617473f20d445d7db908a62dd4a821 --- /dev/null +++ b/99E1T4oBgHgl3EQfCgIZ/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1c1d47a7507325f09afc9c030d9746819b3c1aeeca8062691e108d440bb4a664 +size 204687 diff --git a/99FAT4oBgHgl3EQfqB0k/vector_store/index.faiss b/99FAT4oBgHgl3EQfqB0k/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..30a3f75297de09f619c89d11cac22aa10622ffcd --- /dev/null +++ b/99FAT4oBgHgl3EQfqB0k/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b6d4d07f007c022f5f68387b2c6a04a6f26b30bbea27a1a937d1972062d774d9 +size 6291501 diff --git a/A9AyT4oBgHgl3EQfRvd3/content/tmp_files/2301.00072v1.pdf.txt b/A9AyT4oBgHgl3EQfRvd3/content/tmp_files/2301.00072v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5465b1002e0735a5e88ee9fb6074124294898fed --- /dev/null +++ b/A9AyT4oBgHgl3EQfRvd3/content/tmp_files/2301.00072v1.pdf.txt @@ -0,0 +1,2049 @@ +LeaFTL: A Learning-based Flash Translation Layer +for Solid-State Drives +Jinghan Sun +UIUC +js39@illinois.edu +Shaobo Li +UIUC +shaobol2@illinois.edu +Yunxin Sun∗ +ETH Zurich +yunsun@student.ethz.ch +Chao Sun +Western Digital Research +chao.sun@wdc.com +Dejan Vucinic +Western Digital Research +dejan.vucinic@wdc.com +Jian Huang +UIUC +jianh@illinois.edu +ABSTRACT +In modern solid-state drives (SSDs), the indexing of flash pages is a +critical component in their storage controllers. It not only affects +the data access performance, but also determines the efficiency +of the precious in-device DRAM resource. A variety of address +mapping schemes and optimizations have been proposed. However, +most of them were developed with human-driven heuristics. +In this paper, we present a learning-based flash translation layer +(FTL), named LeaFTL, which learns the address mapping to tolerate +dynamic data access patterns via linear regression at runtime. By +grouping a large set of mapping entries into a learned segment, it +significantly reduces the memory footprint of the address mapping +table, which further benefits the data caching in SSD controllers. +LeaFTL also employs various optimization techniques, including +out-of-band metadata verification to tolerate mispredictions, opti- +mized flash allocation, and dynamic compaction of learned index +segments. We implement LeaFTL with both a validated SSD sim- +ulator and a real open-channel SSD board. Our evaluation with +various storage workloads demonstrates that LeaFTL saves the +memory consumption of the mapping table by 2.9× and improves +the storage performance by 1.4× on average, in comparison with +state-of-the-art FTL schemes. +CCS CONCEPTS +• Hardware → External storage; • Computer systems orga- +nization → Architectures; • Computing methodologies → +Learning linear models. +KEYWORDS +Learning-Based Storage, Flash Translation Layer, Solid-State Drive +1 +INTRODUCTION +Flash-based SSDs have become an indispensable part in modern +storage systems, as they outperform conventional hard-disk drives +(HDDs) by orders of magnitude, and their cost is close to that of +HDDs [22, 30, 51, 62]. The SSD capacity continues to boost by +increasing the number of flash channels and chips with the rapidly +shrinking process and manufacturing technology [22, 25, 41, 46]. +The flash translation layer (FTL) is the core component of man- +aging flash memory in SSDs, including address translation, garbage +collection (GC), and wear leveling [20, 66]. The FTL maintains meta- +data structures for different functions such as address translation +∗Work done when visiting the Systems Platform Research Group at UIUC as a research +intern. +and valid page tracking, and caches them in the in-device DRAM +(SSD DRAM) for improved performance [7, 12, 25]. +Among these data structures, the address mapping table has +the largest memory footprint. In general, the address mapping +table can be categorized in three types: page-level mapping, block- +level mapping, and hybrid mapping. Modern SSDs usually use the +page-level mapping, as it offers the best performance for the flash +page lookup, and incurs minimal GC overhead, in comparison with +the other two mapping schemes [20, 66]. However, the page-level +mapping table size is large, as it stores the entry for the LPA-to-PPA +address translation for each flash page. +The address mapping table significantly affects the performance +of SSDs, as it not only determines the efficiency of indexing flash +pages, but also affects the utilization of SSD DRAM. Moreover, due +to the limitations of the cost and power budget in SSD controllers, +it is challenging for SSD vendors to scale the in-device DRAM +capacity [12, 41]. This challenge becomes even worse with the +increasing flash memory capacity in an SSD, as larger capacity +usually requires a larger address mapping table for indexing. +To improve the address mapping and translation for SSDs, vari- +ous optimization schemes have been developed [9, 25, 29, 38, 39, 66]. +However, most of them were developed based on human-driven +heuristics [25], and cannot capture dynamic data access patterns +at runtime. Employing more semantic knowledge into the FTL, +such as GraphSSD [44], can improve the data indexing and address +translation, however, it is application specific and complicates the +management of address mappings [7], which does not scale for the +development of generic SSDs. In this work, we do not expect that +we can obtain application semantics from the host and the SSD con- +troller. Instead, we focus on utilizing simple yet effective machine +learning (ML) techniques to automate the address mapping table +management in the SSDs, with the capability of learning diverse +and dynamic data access patterns. +To this end, we propose a learning-based FTL, named LeaFTL, by +utilizing the piecewise linear regression technique to learn the LPA- +PPA mappings, and automatically exploiting the data locality of +various data access patterns at runtime. Unlike the state-of-the-art +page-level mapping, the key idea of LeaFTL is that it can learn the +correlation between a set of LPAs and their mapped PPAs, based +on which it can build a space-efficient index segment, as presented +in A in Figure 1. Since the learned index segment can be simply +represented with (𝑆, 𝐿, 𝐾, 𝐼), where [𝑆,𝑆 + 𝐿] denotes the interval +of LPAs, 𝐾 is the slope of the segment, and 𝐼 is the intercept of the +segment (see the last diagram in Figure 1), each segment will take +arXiv:2301.00072v1 [cs.OS] 30 Dec 2022 + +Jinghan Sun, Shaobo Li, Yunxin Sun, Chao Sun, Dejan Vucinic, and Jian Huang +30 +LPA +PPA +31 +32 +33 +34 +155 +156 +157 +158 +159 +60 +62 +64 +66 +68 +200 +201 +203 +204 +205 +80 +82 +83 +84 +87 +304 +305 +306 +307 +308 +Index Segment +A +Index Segment +B +Index Segment +C +LPA +PPA +A +B +C +error bound +1 +1 +1 +1 +2 +2 +2 +2 +2 +1 +1 +3 +Figure 1: An illustrative example of learning LPA-PPA mappings using piecewise linear regression in LeaFTL. It can learn +various patterns of LPA-PPA mappings with guaranteed error bound. Each learned index segment can be represented with +(𝑆, 𝐿, 𝐾, 𝐼), where [𝑆,𝑆 + 𝐿] denotes the interval of LPAs, 𝐾 is the slope, and 𝐼 is the intercept of the index segment. +only 8 bytes (1 byte for 𝑆 and 𝐿, 2 bytes for 𝐾, and 4 bytes for 𝐼) +with our optimizations (see the details in §3). Compared to the on- +demand page-level mapping [20], the learned segment reduces the +mapping table size by a factor of 𝑚 ∗ 𝑎𝑣𝑔(𝐿)/8, where 𝑚 is the size +(8 bytes) of each entry in the on-demand page-level mapping table, +and 𝑎𝑣𝑔(𝐿) is the average number of LPA-PPA mappings that can +be represented in a learned index segment, 𝑎𝑣𝑔(𝐿) is 20.3 according +to our study of various storage workloads. +Beyond learning contiguous LPA-PPA mappings, LeaFTL also +learns different correlation patterns, such as regular and irregular +strided data accesses as shown in B and C , respectively. Unlike +existing indexing optimizations based on human-driven heuristics, +LeaFTL can learn more irregular patterns of LPA-PPA mappings +with guaranteed error bound, as shown in C . This enables LeaFTL +to further condense the address mapping table. Therefore, given a +limited DRAM capacity in the SSD controller, LeaFTL can maximally +utilize the DRAM caching and improve the storage performance. +For the worst case like random I/O accesses, LeaFTL will transfer +the mapping into single-point linear segments (𝐿 = 0, 𝐾 = 0, and +𝐼 = 𝑃𝑃𝐴 in Figure 1), and its memory consumption will be no more +than that of the page-level mapping. +With the learned index segments, LeaFTL may occasionally re- +turn an inaccurate PPA (i.e., address misprediction), which incurs +additional flash accesses until the correct PPA is identified. To over- +come this challenge, we develop an error-tolerant mechanism in +LeaFTL. For each flash page access, we use the reverse mapping +stored in the out-of-band (OOB) metadata of each flash page to +verify the correctness of the data access. Since the OOB usually has +64–256 bytes [20, 23], we use it to store the accurate LPAs mapped +to the neighbor PPAs. Thus, upon an address misprediction, we use +the stored reverse mappings to find the correct PPA, avoiding addi- +tional flash accesses. LeaFTL leverages the intrinsic OOB structure +to handle address mispredictions and make SSD perfectly-suited +for practical learned indexing. +Due to the intrinsic out-of-place write property of SSDs (see +§2), the learned index segments will be disrupted by writes and +GC, and the segments need to be relearned with new LPA-PPA +mappings. To tolerate these disruptions, the learned segments are +organized within multiple levels to maintain the temporal order +in a log-structured manner: the topmost level has the most recent +segments, and the lower level stores older segments. The segments +at the same level are sorted without overlapping. If the new segment +has a conflict with an existing segment, the old segment will be +moved to the lower level. Therefore, LeaFTL can always identify +the latest version of the corresponding LPA-PPA mapping in a top +level of learned index segments. LeaFTL will compact the learned +segments periodically to reduce its memory footprint. +To further maximize the efficiency of LeaFTL, we coordinate its +learning procedure with flash block allocation in the SSD. As flash +block allocation decides the distribution of mapped PPAs, LeaFTL +will allocate consecutive PPAs to contiguous LPAs at its best effort, +for increasing the possibility of learning a space-efficient index seg- +ment. Similar to existing page-level mapping [20, 23], LeaFTL stores +the learned index segments in flash blocks for recovery. Overall, +we make the following contributions: +• We present a learning-based FTL, it can learn various data access +patterns and turn them into index segments for reducing the +storage cost of the mapping table. +• We develop an error-tolerant address translation mechanism to +handle address mispredictions caused by the learned indexes, +with minimal extra flash accesses. +• We preserve the core FTL functions, and enable the coordination +between the learning procedure of the address mapping table +with the flash block allocation and GC to maximize the efficiency +of the learned FTL. +• We manage the learned segments in an optimized log-structured +manner, and enable compaction to further improve the space +efficiency for the address mapping. +We implement LeaFTL with a validated SSD simulator Wisc- +Sim [27] and evaluate its efficiency with a variety of popular storage +workloads. We also develop a system prototype with a real 1TB +open-channel SSD to verify the functions of LeaFTL and validate +its efficiency with real data-intensive applications, such as the key- +value store and transactional database. Our evaluation with the +real SSD shows similar benefits as that of the SSD simulator imple- +mentation. We demonstrate that LeaFTL reduces the storage cost +of the address mapping in the FTL by 2.9× on average. The saved +memory space benefits the utilization of the precious SSD DRAM, +and further improves the storage performance by 1.4× on average. +We also show that LeaFTL does not affect the SSD lifetime, and its + +LeaFTL: A Learning-based Flash-Translation Layer for Solid-State Drives +flash +flash +flash +flash +Flash +Flash +Flash +Flash +DRAM +Flash +Controller +SSD Controller/Firmware +PCIe Interface +Embedded +Processor +Internal Bus +DRAM +Controller +Block I/O +Figure 2: The internal system architecture of SSDs. +learning procedure introduces negligible performance overhead +to the storage processor in the SSD controllers. The codebase of +LeaFTL is available at https://github.com/platformxlab/LeaFTL. +2 +BACKGROUND AND MOTIVATION +Flash-Based Solid-State Drive. An SSD has three major parts +(see Figure 2): a set of flash memory packages, an SSD controller +with embedded processors, and a set of flash controllers. With the +nature of NAND Flash, when a free page is written, the page cannot +be written again until that page is erased. However, erase operation +is performed only at a block granularity. As the erase operation is +expensive, writes are issued to free flash pages erased in advance +(i.e., out-of-place write). GC will be performed to clean the stale +data. As each flash block has limited endurance, it is important for +them to age uniformly (i.e., wear leveling). SSDs have a logical- +to-physical address mapping table to index flash pages. All these +functions are managed by the FTL in the SSD firmware. +Modern SSD controllers have general-purpose embedded pro- +cessors (e.g., ARM processors). The processors help with issuing +I/O requests, translating LPAs to PPAs, and handling GC and wear- +leveling. SSDs also have limited DRAM capacities to cache the +mapping table and the application data. +Address Mapping Table in the FTL. The address mapping table +in FTL generally has three types: page-level mapping, block-level +mapping, and hybrid mapping. The page-level mapping enables di- +rect LPA-PPA mapping for fast lookup. However, each entry usually +takes 8 bytes (4 bytes for LPA, 4 bytes for PPA), and the entire map- +ping table requires large storage space. The block-level mapping +significantly reduces the mapping table size. However, it introduces +additional overhead for the page lookup in the flash block. The hy- +brid mapping takes advantages of both page-level and block-level +mapping. It uses log blocks to store new writes, and index them +with the page-level mapping. The log blocks will be moved into +data blocks that are indexed with block-level mapping. This incurs +significant GC overhead. Therefore, modern SSDs commonly use +the page-level mapping scheme. +Metadata Structures for Flash Management. The FTL usually +employs four metadata structures (see Figure 3): (1) the address +mapping cache ( 1 AMC) for caching the address mapping table +in the SSD DRAM; (2) the global mapping directory ( 2 GMD) for +tracking the locations of the address mapping table pages in the +Address Mapping +Cache (AMC) +1 +Global Mapping +Directory (GMD) +2 +Block Validity +Counter (BVC) +3 +Page Validity +Table (PVT) +4 +LPA +PPA +... +... +LX +PY +... +... +LPA +PPA +... +... +VX +PZ +... +... +PBA +Counter +... +... +... +... +... +... +PBA +Bitmap +... +... +PB +... +... +... +Data Structures in the FTL of Modern SSDs +Flash Memory +Data Blocks +Address Mapping Blocks +Validity Blocks +Figure 3: The common data structures in the FTL of SSDs. +SSD; (3) the block validity counter ( 3 BVC) for tracking the number +of valid pages for each flash block for assisting the GC in the SSD; +and (4) the page validity table ( 4 PVT), which uses bitmaps to +track the valid pages in each flash block. During the GC, the FTL +will check the 3 BVC to select candidate flash blocks, and migrate +their valid pages to free flash blocks. After that, it will erase these +selected flash blocks, and mark them as free blocks. +Limited DRAM Capacity in SSD Controllers. It is hard to provi- +sion large DRAM inside SSD controllers, due to their hardware con- +straints and limited budgets for power and hardware cost [12, 41, 60]. +Thus, SSD controllers often use on-demand caching to maintain +the recently accessed metadata and data in the SSD DRAM. +Among all the metadata structures, the address mapping table +has the largest memory footprint. As discussed, 1 AMC caches the +recently accessed mapping table entries. If a mapping entry is not +cached, the FTL will locate the corresponding address mapping ta- +ble pages stored in the flash blocks, and place the mapping entry in +the 1 AMC. As we scale the SSD capacity, the DRAM challenge will +become even worse. To overcome this challenge, various optimiza- +tions on the mapping table have been proposed [9, 25, 29, 31, 38, 39] +to improve the utilization of the SSD DRAM. However, most of +them cannot automatically capture diverse data access patterns at +runtime, leaving a large room for improvement. +3 +DESIGN AND IMPLEMENTATION +To develop LeaFTL in the SSD controller, we have to overcome the +following research challenges. +• LeaFTL should be able to automatically capture diverse data +access patterns, and generate memory-efficient address mapping +(§3.1, §3.2, §3.3, and §3.4). +• LeaFTL may incur address mispredictions, which could incur +additional flash accesses. LeaFTL should be tolerant of errors and +have low misprediction penalty (§3.5). +• LeaFTL should work coordinately with other core FTL functions +that include GC and wear leveling (§3.6). +• LeaFTL should be lightweight and not incur much extra overhead +to storage operations (§3.7, §3.8 and §3.9). + +Jinghan Sun, Shaobo Li, Yunxin Sun, Chao Sun, Dejan Vucinic, and Jian Huang +(a) Precise Linear Approximation  +(b) Inaccurate Linear Approximation  +Figure 4: Visualization of learned index segments. +1 +2 +4 +8 +16 +32 +64 +128 +256 +512 1024 2048 +Length of Learned Segments +0 +20 +40 +60 +80 +100 +Percentage of +Segments (%) +=0, #Segments=5540 +=4, #Segments=4267 +=8, #Segments=3718 +Figure 5: Aggregated distribution of learned segments. +3.1 +Key Ideas of LeaFTL +Instead of using the space-consuming one-to-one mapping in the +page-level mapping, the key idea of LeaFTL is to exploit learning +techniques to identify various LPA-PPA mapping patterns and build +efficient learned address mapping entries. Modern SSD controllers +usually have a data buffer for grouping writes and write the large +data chunk at once for exploiting the internal flash parallelisms. +LeaFTL utilizes this data buffer to collect LPA-to-PPA mappings for +learning index segments for free, and does not introduce extra data +collection overhead (see the details in §3.3). +As shown in Figure 4 (a), the PPA of an LPA can be obtained +with the expression: 𝑃𝑃𝐴 = 𝑓 (𝐿𝑃𝐴) = ⌈𝐾 ∗ 𝐿𝑃𝐴 + 𝐼⌉, 𝐿𝑃𝐴 ∈ +[𝑆𝐿𝑃𝐴,𝑆𝐿𝑃𝐴 + 𝐿], where [𝑆𝐿𝑃𝐴,𝑆𝐿𝑃𝐴 + 𝐿] denotes the interval (𝐿) +of LPAs, 𝐾 is the slope, and 𝐼 is the intercept. As discussed in §1, +each learned index segment can be represented in 8 bytes: 1 byte for +𝑆𝐿𝑃𝐴 and 𝐿, respectively; 2 bytes for 𝐾, and 4 bytes for 𝐼. The size +of 𝑆𝐿𝑃𝐴 is reduced from 4 bytes to 1 byte with our optimizations +on the segment management (see §3.4). +We can relax the linear regression to capture more flash access +patterns, which further reduces the learned address mapping table +size. As shown in Figure 4 (b), the linear regression can learn a +pattern with guaranteed error bound [−𝛾,𝛾]. As we increase 𝛾, we +can cover more flash access patterns. We applied the relaxed linear +regression with different 𝛾 values to a variety of storage workloads +(see §4.1), our experimental results demonstrate that the number +of learned index segments is gradually decreased, as we increase 𝛾. +Figure 5 shows that 98.2–99.2% of the learned index segments cover +Segment +SLPA +L +K +I +1B +1B +2B +4B +Type +LPAs +PPAs +Index Segment +Accurate +[0, 1, 2, 3] +[32, 33, 34, 35] +Approximate +[0, 1, 4, 5] +[64, 65, 66, 67] +0 +3 +1.00 +32 +0 +5 +0.56 +64 +Figure 6: Types of learned segments in LeaFTL. +up to 128 LPA-PPA mapping entries, demonstrating the potential +advantages of the learning-based approach. +As for random access patterns, LeaFTL will transfer the learned +segments into single-point segments. And these linear segments +do not require more storage space than the page-level mapping. +3.2 +Learned Index Segment +Types of Learned Index Segment. The mapping table of LeaFTL +is built with learned index segments. It has two types of segments: +accurate and approximate segments, as shown in Figure 6. Both of +them are learned with piecewise linear regression technique [64]. +As for the accurate index segments, given an LPA, we can pre- +cisely get the corresponding PPA with 𝑓 (𝐿𝑃𝐴) = ⌈𝐾 ∗ 𝐿𝑃𝐴 + 𝐼⌉. +For example, when the LPA is 2 in Figure 6, we can directly get the +PPA value of 34 with ⌈1.00 ∗ 2 + 32⌉. In this example, the learned +segment has 𝐿 = 3 and it indexes 4 LPA-PPA mappings. If 𝐿 = 0, +the learned segment will become a single-point segment, the slope +𝐾 = 0, and we will get its PPA with 𝑃𝑃𝐴 = 𝐼. +As for approximate index segments, we use the same formula +𝑓 (𝐿𝑃𝐴) = ⌈𝐾 ∗𝐿𝑃𝐴+𝐼⌉ to calculate the PPA. However, the returned +PPA may not be the exact corresponding PPA. It has an error bound +[−𝛾,𝛾] guaranteed by the linear regression, and 𝛾 is configurable. +For example, given 𝐿𝑃𝐴 = 4 in Figure 6, the value of the PPA is +67, according to the calculation ⌈4 ∗ 0.56 + 64⌉. However, the real +PPA should be 66. We define this as address misprediction. We will +discuss how we handle the address misprediction with reduced +miss penalty in §3.5. +Size of Learned Index Segment. As discussed in §3.1, each seg- +ment can be expressed in (𝑆𝐿𝑃𝐴, 𝐿, 𝐾, 𝐼). The starting LPA will take +4 bytes. We can further reduce this size by partitioning a range of +LPAs into small groups, and each LPA group represents a certain +number of contiguous LPAs. Therefore, we can index an LPA with +its offset in a corresponding group. In LeaFTL, each group repre- +sents 256 contiguous LPAs. Thus, 𝑆𝐿𝑃𝐴 can be indexed by the offset +(28 = 256) in the group, which takes only 1 byte. We use 256 as the +group size, because the length of the learned segments is usually +less than 256 (see Figure 5). +Given an LPA, we can get its offset in the group with (𝐿𝑃𝐴 𝑚𝑜𝑑 +256). In LeaFTL, we set the 𝐿 as 1 byte. Thus, each segment can +index 256 LPA-PPA mappings. We use a 16-bit floating point to +store the value of the slope 𝐾. And the intercept 𝐼 of a segment +can be represented in 4 bytes. Therefore, in combination with 𝑆𝐿𝑃𝐴, +both accurate and approximate segments can be encoded with 8 +bytes (see Figure 6), which are memory aligned. + +LeaFTL: A Learning-based Flash-Translation Layer for Solid-State Drives +(a) Unoptimized learned segments +(b) Optimized learned segments with sorting +Learned Segments +78 +32  33 +76 +Flush +Data Buffer +115 +34  38 +Flash Block +78 +32 +33 +76 +115 +34 +38 +... +LPA 78 +32 +33 +76 115 34 +38 +Learned Segments +Flush +Data Buffer +Flash Block +32 +33 +34 +38 +76 +78 +115 +... +LPA 78 +32 +33 +76 115 34 +38 +115 +32  33  34  38 76  78 +Figure 7: An example of reducing the number of learned seg- +ments via exploiting the flash block allocation. +LeaFTL uses the least significant bit of the 𝐾 to indicate segment +types (0 for accurate segments, 1 for approximate segments). This +has negligible impact on the address translation accuracy, because +𝐾 ∈ [0, 1], which will only affect the tenth digit after decimal point. +3.3 +Improve the Learning Efficiency +To further reduce the number of learned segments, LeaFTL performs +optimizations to improve its learning efficiency of address mappings +by exploiting the flash block allocation in SSD controllers, as shown +in Figure 7. Flash pages are usually buffered in the SSD controller +and written to flash chips at a flash block granularity, for utilizing +the internal bandwidth and avoiding the open-block problem [6, +22, 37, 48]. This allows LeaFTL to learn more space-efficient index +segments (i.e., index segments can cover more LPA-PPA mappings) +by reordering the flash pages with their LPAs in the data buffer. +As shown in Figure 7 (a), LeaFTL learns 5 index segments (78), (32, +33), (76), (115), and (34, 38) with 𝛾 = 4. After sorting the pages in +the data buffer shown in Figure 7 (b), LeaFTL generates 3 index +segments (32, 33, 34, 38), (76, 78), and (115). +To develop the optimized learned segments, LeaFTL sorts the +flash pages in ascending order of their LPAs in the data buffer (8MB +by default). When pages in the data buffer is flushed to the flash +chips, their PPAs are in ascending order. This ensures a mono- +tonic address mapping between LPAs and PPAs, which reduces the +number of index segments. +3.4 +Manage Learned Index Segments +Upon new data updates or GC in the SSD, the learned index seg- +ments need to be updated, due to the intrinsic property (i.e., out-of- +place update) of SSDs. Unfortunately, the direct updates to learned +index segments are expensive, since we have to relearn the in- +dex segments with new PPAs. This relearning procedure not only +consumes extra compute cycles, but also involves additional flash +accesses, since we have to access the corresponding flash pages to +obtain accurate PPAs for some of the LPAs in the index segment +being updated. For instance, for in-place update to an approximate +Level 0 +Level 1 +0 63 +100 200 230 255 +16 127 +206 240 +non-overlapping +at each level +segments can overlap +across levels +Figure 8: The learned index segments are managed in a log- +structured manner in LeaFTL. +segment, it can incur 21 flash accesses on average when relearn- +ing. In-place update also breaks the existing LPA-to-PPA mapping +patterns, which results in 1.2× additional segments and memory +footprint, according to our experiments with various workloads. +To address this challenge, we manage the learned index segments +in a log-structured manner, as shown in Figure 8. Therefore, the +newly learned index segments will be appended to the log structure +(level 0 in Figure 8) and used to index the updated LPA-PPA map- +pings, while the existing learned segments (level 1 and lower levels +in Figure 8) can still serve address translations for LPAs whose map- +pings have not been updated. Such a structure supports concurrent +lookups as enabled in the traditional log-structured merge tree. As +we insert the newly learned index segments at the top level of the +log-structured tree, this minimizes the impact on other segments. +Log-Structured Mapping Table. The log-structured mapping ta- +ble has multiple levels to maintain the temporal order of index seg- +ments. As discussed, the topmost level has the most recent learned +index segments, and the lower level stores the older segments. For +the segments on the same level, LeaFTL ensures that they are sorted +and do not have overlapped LPAs. This is for fast location of the +corresponding learned index segments in each level. For the seg- +ments across the levels, they may have overlapped LPAs, due to the +nature of the log-structured organization. And the segments with +overlapped LPA-PPA mappings will be compacted periodically for +space reclamation (see its detailed procedure in §3.7). +Manage Two Types of Index Segments. LeaFTL manages the ac- +curate and approximate index segments in the same log-structured +mapping table, as they can be encoded in the same format. For each +accurate segment, we can directly infer its indexed LPAs with the +𝑆𝐿𝑃𝐴, 𝐾, and 𝐿, since it has a regular pattern. However, for approx- +imate index segments, we only have the knowledge of the starting +LPA and the end LPA with 𝑆𝐿𝑃𝐴 + 𝐿. Its encoded LPAs cannot be +directly inferred from their metadata (𝑆𝐿𝑃𝐴, 𝐿, 𝐾, 𝐼), since they are +learned from irregular access patterns and may have mispredictions. +If two approximate segments have overlapping LPA ranges, we +could obtain inaccurate PPAs from the learned index segments. +As shown in Figure 9 (a), given an LPA with the value 105, we +will check the segment at Level 0 and may get an inaccurate PPA. +This will also affect the efficiency of the segment compaction, with +which we eliminate duplicated entries between segments. +To address this challenge, LeaFTL uses a Conflict Resolution +Buffer (CRB) for each LPA group to store the LPAs indexed by each +approximate segment. The main purpose of CRB is to help LeaFTL +check whether a given LPA belongs to one approximate segment. +The CRB is a nearly-sorted list [10] by the starting LPAs of its ap- +proximate segments. To be specific, the CRB ensures the following + +Jinghan Sun, Shaobo Li, Yunxin Sun, Chao Sun, Dejan Vucinic, and Jian Huang +100 +6 +K1 +I1 + [100, 101, 103, 104, 106] +102 +6 +K2 +I2 +[102, 105, 107, 108] +L0 +L1 +LPAs +Lookup (LPA = 105) +(a) Approximate index segments that index overlapped LPAs. +Conflict Resolution Buffer +100 +101 +103 +104 +106 +null +102 +105 +107 108 +null +... +Lookup (LPA = 105) +102 +6 +K2 +I2 +(b) Resolve the conflict between approximate segments with CRB +Figure 9: A case study of conflict resolution buffer for ap- +proximate learned index segments. +properties: (1) the LPAs belong to the same approximate segment +are stored contiguously; (2) different approximate segments are +sorted by their starting LPA, and CRB uses a 𝑛𝑢𝑙𝑙 byte to separate +these segments; (3) it does not have redundant LPAs, which means +an LPA will appear at most once in the CRB. This is achieved by +removing existing same LPAs when we insert new approximate +segments into the CRB. +However, if the 𝑆𝐿𝑃𝐴 of a new approximate segment is the same +as any starting LPAs that have been stored in the CRB, LeaFTL will +update the 𝑆𝐿𝑃𝐴 of the old segment with the adjacent LPA. Take +Figure 9 (b) as an example, upon a new approximate segment with +𝑆𝐿𝑃𝐴 = 100, we will update the 𝑆𝐿𝑃𝐴 of the existing segment to 101, +and then insert the new segment into the CRB. In this case, LeaFTL +will ensure each approximate segment will have its unique 𝑆𝐿𝑃𝐴. +This will facilitate the approximate LPA-PPA address translation +with high accuracy confidence. +Since CRB is nearly sorted, its insertion, deletion, and lookup +operations are fast. The CRB is also space efficient, as each LPA +(the offset in its corresponding LPA group) will take only one byte, +and it guarantees that there are no redundant LPAs. Therefore, the +CRB will maximally store 256 LPAs. Our experiments with a variety +of storage workloads show that the CRB will take 13.9 bytes on +average, as shown in Figure 10. +Given an LPA, in order to identify which approximate index +segment it belongs to, LeaFTL will check the CRB with binary +search. Once the LPA is found, LeaFTL will search to its left until +identifying the 𝑆𝐿𝑃𝐴, and this 𝑆𝐿𝑃𝐴 will be the starting LPA of +the corresponding approximate segment, as shown in Figure 9 (b). +Therefore, CRB can assist LeaFTL to resolve the LPA lookups. +3.5 +Handle Address Misprediction +As discussed in §3.2, the mapping table entries encoded with ap- +proximate segments may occasionally incur mispredictions and +return an approximated PPA. These approximate segments have a +guaranteed error bound [−𝛾,𝛾], where 𝛾 is a constant value that +can be specified in the linear regression algorithm. To verify the +correctness of the address translation, a simple method is to access +MSR-hm +MSR-src2 +MSR-prxy +MSR-prn +MSR-usr +FIU-home +FIU-mail +0 +100 +200 +300 +CRB Size (in Bytes) +Average +99 Percentile +Figure 10: The distribution of CRB sizes for different storage +workloads, when we set 𝛾 = 4 in LeaFTL. +PPA1 +PPA2 +PPA3 +PPA4 +PPA5 +Data Blocks +Data +OOB +Flash Page +LPA2 +LPA4 +LPA +Reverse Mapping +Figure 11: The out-of-band (OOB) metadata organization. It +stores the reverse mapping for its neighbor PPAs. +the flash page with the predicted PPA, and use the reverse mapping +(its corresponding LPA) stored in the OOB metadata of the flash +page to check whether the LPA matches or not. In this case, upon +a PPA misprediction, we need log(𝛾) flash accesses on average to +identify the correct PPA. +To avoid extra flash accesses for address mispredictions, LeaFTL +leverages the OOB of the flash page to store the reverse mappings +of its neighbor PPAs. This is developed based on the insight that: +with a 𝑃𝑃𝐴𝑙𝑒𝑎𝑟𝑛𝑒𝑑 obtained from an approximate segment, its er- +ror bound [−𝛾,𝛾] guarantees that the correct PPA is in the range +of [𝑃𝑃𝐴𝑙𝑒𝑎𝑟𝑛𝑒𝑑 − 𝛾, 𝑃𝑃𝐴𝑙𝑒𝑎𝑟𝑛𝑒𝑑 + 𝛾], as discussed in Figure 4 (b). +Thus, upon a misprediction, LeaFTL will read the flash page with +𝑃𝑃𝐴𝑙𝑒𝑎𝑟𝑛𝑒𝑑, and use its OOB to find the correct PPA. In this case, +LeaFTL ensures that it will incur only one extra flash access for +address mispredictions. +This is a feasible approach, as the OOB size is usually 128–256 +bytes in modern SSDs. As each LPA takes 4 bytes, we can store +32–64 reverse mapping entries in the OOB. We show the OOB +organization of LeaFTL in Figure 11. For the flash page 𝑃𝑃𝐴𝑋 , the +first 2𝛾 + 1 entries in its OOB correspond to the LPAs for the flash +pages [𝑃𝑃𝐴𝑋 − 𝛾, 𝑃𝑃𝐴𝑋 + 𝛾]. For the flash pages at the beginning +and end of a flash block, we may not be able to obtain the reverse +mapping of their neighbor PPAs. We place the 𝑛𝑢𝑙𝑙 bytes in the +corresponding entry of the OOB. +3.6 +Preserve Other Core FTL Functions +LeaFTL preserves the core functions such as GC and wear leveling +in an FTL. It follows the same GC and wear leveling policies in +modern SSDs. When the number of free blocks in an SSD is below +a threshold (usually 15-40% of the total flash blocks), the SSD con- +troller will trigger the GC execution. LeaFTL employs the greedy +algorithm [5] to select the candidate blocks which have the minimal + +LeaFTL: A Learning-based Flash-Translation Layer for Solid-State Drives +ALGORITHM 1: LeaFTL operations +Input: 𝑔𝑟𝑜𝑢𝑝𝑠 ← 𝐿𝑒𝑎𝐹𝑇𝐿 𝑔𝑟𝑜𝑢𝑝 𝑝𝑎𝑟𝑡𝑖𝑡𝑖𝑜𝑛𝑠 +// Insert/Update Segment in the LeaFTL +1 Function 𝑠𝑒𝑔_𝑢𝑝𝑑𝑎𝑡𝑒(𝑠𝑒𝑔𝑚𝑒𝑛𝑡,𝑙𝑒𝑣𝑒𝑙): +2 +𝑠𝑒𝑔_𝑝𝑜𝑠 = 𝑏𝑖𝑛𝑎𝑟𝑦_𝑠𝑒𝑎𝑟𝑐ℎ(𝑙𝑒𝑣𝑒𝑙,𝑠𝑒𝑔𝑚𝑒𝑛𝑡.𝑆𝐿𝑃𝐴) +3 +𝑙𝑒𝑣𝑒𝑙.𝑖𝑛𝑠𝑒𝑟𝑡 (𝑠𝑒𝑔𝑚𝑒𝑛𝑡,𝑠𝑒𝑔_𝑝𝑜𝑠) +4 +if 𝑛𝑜𝑡 𝑠𝑒𝑔𝑚𝑒𝑛𝑡.𝑎𝑐𝑐𝑢𝑟𝑎𝑡𝑒 then +5 +Insert LPAs into CRB and remove redundant LPAs +6 +if 𝑠𝑒𝑔𝑚𝑒𝑛𝑡.𝑆𝐿𝑃𝐴 exists in CRB then +7 +Update the 𝑆𝐿𝑃𝐴 of the old segment +8 +𝑣𝑖𝑐𝑡𝑖𝑚_𝑠𝑒𝑔𝑚𝑒𝑛𝑡𝑠 ← All segments that overlap the 𝑠𝑒𝑔𝑚𝑒𝑛𝑡 +starting with 𝑠𝑒𝑔_𝑝𝑜𝑠 +9 +foreach 𝑣𝑖𝑐𝑡𝑖𝑚 ∈ 𝑣𝑖𝑐𝑡𝑖𝑚_𝑠𝑒𝑔𝑚𝑒𝑛𝑡𝑠 do +10 +𝑠𝑒𝑔_𝑚𝑒𝑟𝑔𝑒 (𝑠𝑒𝑔𝑚𝑒𝑛𝑡, 𝑣𝑖𝑐𝑡𝑖𝑚) +// if marked as removable by seg_merge() +11 +if 𝑣𝑖𝑐𝑡𝑖𝑚.𝐿 = −1 then +12 +𝑙𝑒𝑣𝑒𝑙.𝑟𝑒𝑚𝑜𝑣𝑒 (𝑣𝑖𝑐𝑡𝑖𝑚) +13 +if 𝑠𝑒𝑔𝑚𝑒𝑛𝑡.𝑜𝑣𝑒𝑟𝑙𝑎𝑝𝑠 (𝑣𝑖𝑐𝑡𝑖𝑚) then +14 +Pop 𝑣𝑖𝑐𝑡𝑖𝑚 to the next level +15 +if 𝑣𝑖𝑐𝑡𝑖𝑚 has overlaps in the next level then +16 +Create level for 𝑣𝑖𝑐𝑡𝑖𝑚 to avoid recursion +// Lookup LPA in the LeaFTL +17 Function 𝑙𝑜𝑜𝑘𝑢𝑝(𝑙𝑝𝑎): +18 +foreach 𝑙𝑒𝑣𝑒𝑙 ∈ 𝑔𝑟𝑜𝑢𝑝𝑠 [𝑙𝑝𝑎 𝑚𝑜𝑑 256] do +19 +𝑠𝑒𝑔_𝑝𝑜𝑠 = 𝑏𝑖𝑛𝑎𝑟𝑦_𝑠𝑒𝑎𝑟𝑐ℎ(𝑙𝑒𝑣𝑒𝑙,𝑙𝑝𝑎) +20 +𝑠𝑒𝑔𝑚𝑒𝑛𝑡 = 𝑙𝑒𝑣𝑒𝑙.𝑔𝑒𝑡_𝑠𝑒𝑔𝑚𝑒𝑛𝑡 (𝑠𝑒𝑔_𝑝𝑜𝑠) +21 +if ℎ𝑎𝑠_𝑙𝑝𝑎(𝑠𝑒𝑔𝑚𝑒𝑛𝑡, 𝑙𝑝𝑎) then +22 +return 𝑠𝑒𝑔𝑚𝑒𝑛𝑡.𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑒𝑃𝑃𝐴(𝑙𝑝𝑎) +// LeaFTL Compaction +23 Function 𝑠𝑒𝑔_𝑐𝑜𝑚𝑝𝑎𝑐𝑡(): +24 +foreach 𝑔𝑟𝑜𝑢𝑝 ∈ 𝑔𝑟𝑜𝑢𝑝𝑠 do +25 +foreach 𝑢𝑝𝑝𝑒𝑟_𝑙𝑒𝑣𝑒𝑙,𝑙𝑜𝑤𝑒𝑟_𝑙𝑒𝑣𝑒𝑙 ∈ 𝑔𝑟𝑜𝑢𝑝 do +26 +foreach 𝑠𝑒𝑔𝑚𝑒𝑛𝑡 ∈ 𝑢𝑝𝑝𝑒𝑟_𝑙𝑒𝑣𝑒𝑙 do +27 +𝑠𝑒𝑔_𝑢𝑝𝑑𝑎𝑡𝑒 (𝑠𝑒𝑔𝑚𝑒𝑛𝑡,𝑙𝑜𝑤𝑒𝑟_𝑙𝑒𝑣𝑒𝑙) +28 +if 𝑢𝑝𝑝𝑒𝑟_𝑙𝑒𝑣𝑒𝑙 is empty then +29 +𝑔𝑟𝑜𝑢𝑝.𝑟𝑒𝑚𝑜𝑣𝑒 (𝑢𝑝𝑝𝑒𝑟_𝑙𝑒𝑣𝑒𝑙) +number of valid pages, for reducing the data movement overhead +at GC. As the GC move the valid pages from the candidate blocks +to the free blocks, LeaFTL places these valid pages into the DRAM +buffer, sort them by their LPAs, and learn a new index segment. +The learning procedure is the same as we build index segments for +new flash writes/updates. Thus, the address mapping of the valid +pages is updated after the GC. +LeaFTL also ensures all the flash blocks age at the same rate +(i.e., wear leveling). It uses the throttling and swapping mechanism +developed in existing GC, in which the cold data blocks (i.e., blocks +not frequently accessed) will be migrated to hot blocks (i.e., blocks +that experience more wear). LeaFTL will learn new indexes for +these swapped blocks and insert them into the mapping table to +update their address mappings. +3.7 +LeaFTL Operations +Now we describe the LeaFTL operations, including segment cre- +ation, insert/update, LPA lookup, and compaction. We discuss their +procedures, and use examples to illustrate each of them, respec- +tively. We present their detailed procedures in Algorithm 1 and 2. +ALGORITHM 2: Segment Merge +// Check if Segment Contains LPA +1 Function ℎ𝑎𝑠_𝑙𝑝𝑎(𝑠𝑒𝑔, 𝑙𝑝𝑎): +2 +𝑎𝑐𝑐 ← 𝑠𝑒𝑔.𝑎𝑐𝑐𝑢𝑟𝑎𝑡𝑒 +3 +if 𝑙𝑝𝑎 ∉ [𝑠𝑒𝑔.𝑆𝐿𝑃𝐴,𝑠𝑒𝑔.𝑆𝐿𝑃𝐴 + 𝑠𝑒𝑔.𝐿] 𝑜𝑟 +(𝑛𝑜𝑡 𝑎𝑐𝑐 & 𝑐ℎ𝑒𝑐𝑘 (𝐶𝑅𝐵) 𝑓 𝑎𝑖𝑙𝑒𝑑) 𝑜𝑟 +(𝑎𝑐𝑐 & (𝑙𝑝𝑎 − 𝑠𝑒𝑔.𝑆𝐿𝑃𝐴) 𝑚𝑜𝑑 ⌈ +1 +𝑠𝑒𝑔.𝐾 ⌉ ≠ 0) then +4 +𝑟𝑒𝑡𝑢𝑟𝑛 𝐹𝑎𝑙𝑠𝑒 +5 +𝑟𝑒𝑡𝑢𝑟𝑛 𝑇𝑟𝑢𝑒 +// Convert Segment into a Temporary Bitmap +6 Function 𝑔𝑒𝑡_𝑏𝑖𝑡𝑚𝑎𝑝(𝑠𝑒𝑔, 𝑠𝑡𝑎𝑟𝑡, 𝑒𝑛𝑑): +7 +𝑏𝑚 ← 𝑏𝑖𝑡𝑚𝑎𝑝 𝑜𝑓 𝑙𝑒𝑛𝑔𝑡ℎ (𝑒𝑛𝑑 − 𝑠𝑡𝑎𝑟𝑡 + 1) +8 +foreach 𝑙𝑝𝑎 ∈ [𝑠𝑡𝑎𝑟𝑡,𝑒𝑛𝑑] do +9 +if ℎ𝑎𝑠_𝑙𝑝𝑎(𝑠𝑒𝑔, 𝑙𝑝𝑎) then +10 +𝑏𝑚[𝑙𝑝𝑎 − 𝑠𝑡𝑎𝑟𝑡 ] = 1 +11 +else +12 +𝑏𝑚[𝑙𝑝𝑎 − 𝑠𝑡𝑎𝑟𝑡 ] = 0 +13 +return 𝑏𝑚 +// Merge a New Segment with an Old Segment +14 Function 𝑠𝑒𝑔_𝑚𝑒𝑟𝑔𝑒(𝑛𝑒𝑤, 𝑜𝑙𝑑): +15 +𝑠𝑡𝑎𝑟𝑡 ← 𝑚𝑖𝑛(𝑛𝑒𝑤.𝑆𝐿𝑃𝐴, 𝑜𝑙𝑑.𝑆𝐿𝑃𝐴) +16 +𝑒𝑛𝑑 ← 𝑚𝑎𝑥 (𝑛𝑒𝑤.𝑆𝐿𝑃𝐴 + 𝑛𝑒𝑤.𝐿, 𝑜𝑙𝑑.𝑆𝐿𝑃𝐴 + 𝑜𝑙𝑑.𝐿) +17 +𝑏𝑚𝑛𝑒𝑤 ← 𝑔𝑒𝑡_𝑏𝑖𝑡𝑚𝑎𝑝 (𝑛𝑒𝑤, 𝑠𝑡𝑎𝑟𝑡, 𝑒𝑛𝑑) +18 +𝑏𝑚𝑜𝑙𝑑 ← 𝑔𝑒𝑡_𝑏𝑖𝑡𝑚𝑎𝑝 (𝑜𝑙𝑑, 𝑠𝑡𝑎𝑟𝑡, 𝑒𝑛𝑑) +19 +𝑏𝑚𝑜𝑙𝑑 ← 𝑏𝑚𝑜𝑙𝑑 & ¬𝑏𝑚𝑛𝑒𝑤 +20 +𝑓 𝑖𝑟𝑠𝑡, 𝑙𝑎𝑠𝑡 ← the first and last valid bit of 𝑏𝑚𝑜𝑙𝑑 +21 +𝑜𝑙𝑑.𝑆𝐿𝑃𝐴, 𝑜𝑙𝑑.𝐿 ← 𝑓 𝑖𝑟𝑠𝑡 + 𝑠𝑡𝑎𝑟𝑡, 𝑙𝑎𝑠𝑡 − 𝑓 𝑖𝑟𝑠𝑡 +22 +if no valid bits in 𝑜𝑙𝑑 then +23 +𝑜𝑙𝑑.𝐿 ← −1 +// mark it as removable +24 +if 𝑛𝑜𝑡 𝑜𝑙𝑑.𝑎𝑐𝑐𝑢𝑟𝑎𝑡𝑒 then +25 +Remove outdated LPAs in CRB +Creation of Learned Segments. Once the data buffer of the SSD +controller is filled, LeaFTL takes the LPAs and PPAs of the flash +pages in the buffer as the input. It sorts the LPA-PPA mappings +by reordering the flash pages with their LPAs (see §3.3), and uses +greedy piecewise linear regression [64] to learn the index segment. +Insert/Update of Learned Segments. When we insert or update +a new learned index segment, we will place it in the topmost level +of the log-structured mapping table. Since each level of the map- +ping table is sorted, we can quickly identify its insert location via +a binary search (line 2 in Algorithm 1). If the new segment is ap- +proximate, LeaFTL will update the CRB for future lookups (line +4-7 in Algorithm 1). After that, LeaFTL will check whether the +new segment overlaps with existing segments. If yes, LeaFTL will +identify the overlapped LPAs. The overlap detection is performed +by the comparison between the LPA range of the new segment and +[𝑆𝐿𝑃𝐴,𝑆𝐿𝑃𝐴 +𝐿] of the adjacent segments. We group these overlap- +ping segments as a list of victim segments (line 8 in Algorithm 1). +LeaFTL will merge segments to remove outdated LPAs (line 10 in +Algorithm 1 and line 14-25 in Algorithm 2). +To fulfill the segment merge, LeaFTL will use the 𝑆𝐿𝑃𝐴, 𝐿, and 𝐾 +to reconstruct the list of the encoded LPAs in the victim segment. +And it will create a bitmap to index these encoded LPAs (line 6-13 +in Algorithm 2). Given an accurate segment with 𝑆𝐿𝑃𝐴 = 100, 𝐾 = +0.5, 𝐿 = 6, we can infer that its encoded LPAs are [100, 102, 104, 106]. +We can transfer the LPA list to the bitmap [1010101]. If the victim + +Jinghan Sun, Shaobo Li, Yunxin Sun, Chao Sun, Dejan Vucinic, and Jian Huang +MSR-hm +MSR-src2 +MSR-prxy +MSR-prn +MSR-usr +FIU-home +FIU-mail +0 +5 +10 +15 +20 +# of Levels +in Each Group +Average +99 Percentile +Figure 12: A study of the number of levels in the log- +structured mapping table for different storage workloads. +L0 +0      63 +T0 +Initial Snapshot +T1 +Update LPAs 200 - 255 +L0 +0     63 +200  255 +T2 +Update LPAs 16 - 31 +L0 +16    31 +200  255 +L1 +0      63 +T4 +Update [72, 73, 80] +L0 +16    31 +200  255 +L1 +0      63 +T6 +Lookup LPA 78 +L0 +L1 +T8 +Compaction +Timeline +Segments +CRB +T7 +Update LPAs 32 - 90 +75     82 +72     80 +16    31 +200  255 +0      63 +75     82 +72     80 +T5 +Lookup LPA 50 +L0 +L1 +16    31 +200  255 +0      63 +75     82 +72     80 +L0 +L1 +16    31 +200  255 +0      63 +75     82 +32     90 +L0 +16   31 +200  255 +0    15 +32   90 +Start      End +Accurate Segment +Start      End +Approximate Segment +72 73 80 +/ 75 78 82 +72 73 80 +/ +75 78 82 +72 73 80 +/ +75 78 82 +75 78 82 +T3 +Update [75, 78, 82] +L0 +16    31 +200  255 +L1 +0      63 +75     82 +75 78 82 +Figure 13: Examples that involve update/insert, lookup, and +compaction operations in LeaFTL. +segment is an approximate segment, LeaFTL will leverage the 𝑆𝐿𝑃𝐴, +𝐿, and the LPAs stored in the CRB to reconstruct the encoded LPAs. +Afterwards, LeaFTL will conduct a comparison between the bitmaps +to identify the overlapped LPAs (line 15-19 in Algorithm 2). +During the segment merge, LeaFTL will update the 𝑆𝐿𝑃𝐴 and 𝐿 +of the old segments accordingly, remove the outdated LPAs from +CRB for approximate segments. Note that we do not update the 𝐾 +and 𝐼 for the victim segments during the merge. +After the merge, (1) if the victim segment does not contain any +valid LPA (𝐿 is negative), it will be removed from the mapping +table (line 11-12 in Algorithm 1). (2) If the victim segment has +valid LPAs but their range still overlaps with the new segment, +the victim segment will be moved to the next level in the log- +structured mapping table (line 13-16 in Algorithm 1). To avoid +recursive updates across the levels, we create a new level for the +victim segment if it also overlaps with segments in the next level. +According to our study of diverse workloads, this will not create +many levels in the mapping table (see Figure 12). (3) If the victim +segment has valid LPAs and they do not overlap with the new +segment, we do not need to perform further operations. This is +because the victim segment is updated with new 𝑆𝐿𝑃𝐴 and 𝐿 during +segment merge (line 20-25 in Algorithm 2), and the new segment +insertion keeps each level sorted (line 3 in Algorithm 1). +To facilitate our discussion, we present a few examples in Fig- +ure 13. At the initial stage, the mapping table has one segment that +indexes the LPA range [0, 63]. At 𝑇1, the new segment [200, 255] is +directly inserted into the topmost level, as it does not overlap with +existing segments. At 𝑇2, we insert a new segment [16, 31] that has +overlaps with the old segment [0, 63], LeaFTL conducts the segment +merge procedure. After that, the old segment still has valid LPAs. +Thus, it moves to level 1. At 𝑇3 and 𝑇4, we insert two approximate +segments [75, 82] and [72, 80], LeaFTL will also insert their encoded +LPAs into the CRB. The segment [75, 82] will be moved to the next +level as it overlaps with the new segment [72, 80]. +LPA Lookup. LeaFTL conducts an LPA lookup from the top- +most level of the mapping table with binary searches (line 19 in +Algorithm 1). We will check whether the LPA is represented by the +matched segment (line 21 in Algorithm 1, line 1-5 in Algorithm 2). If +the 𝐿𝑃𝐴 ∈ [𝑆𝐿𝑃𝐴,𝑆𝐿𝑃𝐴 + 𝐿] of the segment, LeaFTL will check the +least bit of its 𝐾. If the least bit of 𝐾 is 0, it is an accurate segment, +and LeaFTL will use 𝑓 (𝐿𝑃𝐴) = ⌈𝐾 ∗ 𝐿𝑃𝐴 + 𝐼⌉ to get the accurate +PPA (see §3.2). Otherwise, it is an approximate segment. LeaFTL +will check the CRB to identify the 𝑆𝐿𝑃𝐴 of the segment, following +the approach described in Figure 9 and §3.4. LeaFTL will use the +same 𝑓 (𝐿𝑃𝐴) formula to obtain the PPA. If the LPA is not found in +the top level of the mapping table, LeaFTL will search the lower +levels until a segment is identified. +We use Figure 13 to illustrate the lookup procedure. At 𝑇5, we +conduct the address translation for 𝐿𝑃𝐴 = 50. However, none of +the segments in the level 0 covers this LPA, LeaFTL will continue +the search in the level 1 and find the accurate segment [0, 63]. At +𝑇6, we do the address translation for 𝐿𝑃𝐴 = 78. LeaFTL finds that +the LPA 78 is in the LPA range of the segment [72, 80]. Since this +is an approximate segment, LeaFTL checks the CRB and finds this +LPA is actually indexed by the segment [75, 82]. +With the PPA, LeaFTL will read the corresponding flash page and +use the reversed mapping (its corresponding LPA) in its OOB to ver- +ify the correctness of the address translation. Upon mispredictions, +we will use the approach discussed in §3.5 to handle it. +Segment Compaction. The purpose of the compaction is to +merge segments with overlapped LPAs across different levels, which +further saves memory space. LeaFTL will iteratively move the upper- +level segments into the lower level, until the mapping table is fully +compacted (line 27 in Algorithm 1). When an approximate segment +is removed, its corresponding CRB entries will also be deleted. As +shown in 𝑇7 of Figure 13, we insert a new segment [32, 90] which +fully covers the LPA range of the segment [72, 80]. After merge, +LeaFTL removes the old segment [72, 80]. However, some segments + +LeaFTL: A Learning-based Flash-Translation Layer for Solid-State Drives +Conflict Resolution +Buffer (CRB) +Key Data Structures in LeaFTL +6 +Log-Structured +Mapping Table +5 +L0 +L1 +L2 +... +Group +0 +... +CRB +... +... +0 63 +... +16 31 +... +... +64 95 +Figure 14: Key data structures used in LeaFTL. +in the level 0 still overlap with the segments in the level 1. After 𝑇8, +LeaFTL will remove outdated segments and LPAs. +LeaFTL performs segment compaction after each 1 million writes +by default. According to our experiments with various storage work- +loads, the segment compaction of the entire mapping table will take +4.1 milliseconds (the time of 20-40 flash writes) on average. Consider +the low frequency (i.e., once per 1 million writes), the compaction +incurs trivial performance overhead to storage operations. +3.8 +Put It All Together +LeaFTL is compatible with existing FTL implementations. As shown +in Figure 14, it uses the log-structured mapping table ( 5 ) to replace +the address mapping cache ( 1 in Figure 3), and employs CRB ( 6 ) +for assisting the address translation of approximate segments. The +CRB requires trivial storage space in the SSD DRAM (see Figure 10). +Read Operation. For a read request, LeaFTL will first check the +data cache. For a cache hit, LeaFTL serves the read request with +the cached flash page. Otherwise, LeaFTL will perform address +translation with 5 (see §3.7). If there is a misprediction of PPA, +LeaFTL checks the OOB of the mispredicted flash page, read the +correct page (§3.5), and updates the data cache with the page. +Write Operation. For a write request, LeaFTL buffers it in the +data cache. Once the buffered writes reach the size of a flash block, +LeaFTL will allocate a free block. It will sort the writes in the buffer +based on their LPAs, and learn new index segments with the PPAs +of the allocated flash block. This enables LeaFTL to group more LPA- +PPA mappings in the same index segment. After that, LeaFTL will +insert the new index segment in the mapping table, and flush the +buffered data to the flash blocks. For those writes, LeaFTL will also +check whether their LPAs exist in the mapping table. If yes, LeaFTL +will update their corresponding entries in 3 BVC and 4 PVT to +indicate that they become invalid and can be garbage collected in +the future. Otherwise, the new learned segments will have their +LPA-PPA mappings for future address translations. +LeaFTL caches the mapping table in SSD DRAM for fast lookup. +The table will also be stored in the flash blocks. LeaFTL utilizes the +existing 2 GMD to index the translation pages. If a segment is not +found in the cached mapping table, LeaFTL will fetch it from the +translation blocks and place it in the cached mapping table. +Crash Consistency and Recovery. Upon system crashes or power +failures, LeaFTL guarantees the crash consistency of learned in- +dexes. In order to ensure the data durability of DRAM buffer in +SSD controllers, modern SSDs today have employed battery-backed +DRAM and power loss protection mechanisms [1, 2]. With battery- +backed DRAM, LeaFTL has sufficient time to persist the up-to-date +mapping table to the flash blocks and record their PPAs in the GMD +Table 1: SSD configurations in our simulator. +Parameter +Value +Parameter +Value +Capacity +2TB +#Channels +16 +Page size +4KB +OOB size +128B +DRAM size +1GB +Pages/block +256 +Read latency +20𝜇s +Write latency +200𝜇s +Erase +1.5 millisecs +Overprovisioning ratio +20% +( 2 in Figure 3). During the data recovery, LeaFTL reads the GMD +to locate its mapping table and place it into the DRAM. +Without battery-backed DRAM, LeaFTL periodically flushes the +learned mapping table and the Block Validity Counter ( 3 BVC in +Figure 3) into the flash blocks. When GC is triggered, LeaFTL also +flushes the updated mapping table and BVC into the flash blocks. +Upon crashes, LeaFTL will scan all the flash blocks at the channel- +level parallelism, and reconstruct an up-to-date BVC. LeaFTL is able +to identify the flash blocks allocated since the last mapping table +flush, by comparing the up-to-date BVC with the stored BVC in the +SSD. Therefore, LeaFTL only needs to relearn the index segments +for these recently allocated flash blocks and add them into the +mapping table (see §3.4). +3.9 +Implementation Details +SSD Simulator. We implement LeaFTL based on a trace-driven +simulator WiscSim [27], which has provided an event simulation +environment for the end-to-end performance analysis of SSDs. We +extend WiscSim by implementing an LRU-based read-write cache. +LeaFTL also preserves the functions of existing FTL, such as GC and +wear-leveling. To support the learned indexing, LeaFTL employs +a simple linear regression algorithm [65], which incurs negligible +computation overhead with modern storage processors (see §4.5). +The error bound 𝛾 for learned segments is configurable, and we set +it to 0 by default in LeaFTL. +SSD Prototype. We also develop a real system prototype with +an open-channel SSD to validate the functions and efficiency of +LeaFTL. The SSD has 1TB storage capacity with 16 KB flash page +size. It has 16 channels, each channel has 16K flash blocks, and each +flash block has 256 pages. It enables developers to implement their +own FTL in the host by providing basic I/O commands such as read, +write, and erase. We implement LeaFTL with 4,016 lines of code +using C programming language with the SDK library of the device. +4 +EVALUATION +Our evaluation shows that: (1) LeaFTL significantly reduces the +address mapping table size, and the saved memory brings perfor- +mance benefits (§4.2); (2) the benefits of LeaFTL are validated on a +real SSD device (§4.3); (3) LeaFTL can achieve additional memory +savings and performance benefits with larger error-tolerance, and +it demonstrate generality for different SSD configurations (§4.4); +(4) Its learning procedure does not introduce much extra overhead +to the SSD controller (§4.5); (5) It has minimal negative impact on +the SSD lifetime (§4.6). + +Jinghan Sun, Shaobo Li, Yunxin Sun, Chao Sun, Dejan Vucinic, and Jian Huang +Table 2: Real workloads used in our real SSD evaluation. +Workload +Description +OLTP [59] +Transactional benchmark in the FileBench. +CompFlow (CompF) [59] +File accesses in a computation flow. +TPCC [13] +Online transaction queries in warehouses. +AuctionMark (AMark) [13] +Activity queries in an auction site. +SEATS [13] +Airline ticketing system queries. +MSR-hm +MSR-src2 +MSR-prxy +MSR-prn +MSR-usr +FIU-home +FIU-mail +50x +20x +10x +5x +2x +1x +Memory Footprint +Reduction +DFTL +SFTL +LeaFTL +Figure 15: The reduction on the mapping table size of +LeaFTL, in comparison with DFTL and SFTL. +4.1 +Experiment Setup +We examine the efficiency of LeaFTL with both the SSD simula- +tor and real SSD prototype. As for the evaluation with the SSD +simulator, we configure a 2TB SSD with 4KB flash pages and 1GB +DRAM in the SSD controller. We list the core SSD parameters in +Table 1. For other parameters, we use the default setting in the +WiscSim. We use a variety of storage workloads that include the +block I/O traces from enterprise servers from Microsoft Research +Cambridge [45] and workload traces from computers at FIU [16]. +As for the evaluation with the real SSD prototype (see §3.9), we +validate the benefits of LeaFTL using a set of real-world file system +benchmarks and data intensive applications as shown in Table 2. +Before we measure the performance, we run a set of workloads +consisting of various real-world and synthetic storage workload +traces to warm up the SSD and make sure the GC will be executed +during the experiments. +We compare LeaFTL with state-of-the-art page-level mapping +schemes described as follows 1. +• DFTL (Demand-based FTL) [20]: it uses a page-level mapping +scheme, and caches the most recently used address translation +entries in the SSD DRAM. +• SFTL (Spatial-locality-aware FTL) [25]: it is a page-level map- +ping that exploits the spatial locality and strictly sequential access +patterns of workloads to condense mapping table entries. +4.2 +Memory Saving and Performance +We first evaluate the benefits of LeaFTL on the memory saving +and storage performance with the SSD simulator. As shown in +Figure 15, LeaFTL reduces the mapping table size by 7.5–37.7×, +compared to the page-level mapping scheme DFTL. This is because +LeaFTL can group a set of page-level mapping entries into an 8- +byte segment. In comparison with SFTL, LeaFTL achieves up to +5.3× (2.9× on average) reduction on the address mapping table for +different storage workloads, when we set its 𝛾 = 0 (i.e., the learned +1We do not compare LeaFTL with block-level and hybrid-level mappings, as they +perform dramatically worse than the page-level mapping [20, 25]. +MSR-hm +MSR-src2 +MSR-prxy +MSR-prn +MSR-usr +FIU-home +FIU-mail +0.0 +0.5 +1.0 +Normalized Perf. +DFTL +SFTL +LeaFTL +(a) SSD performance when using its DRAM mainly for the address +mapping table (lower is better). +MSR-hm +MSR-src2 +MSR-prxy +MSR-prn +MSR-usr +FIU-home +FIU-mail +0.0 +0.5 +1.0 +Normalized Perf. +DFTL +SFTL +LeaFTL +(b) SSD performance when using its DRAM partially (up to 80%) for +the address mapping table (lower is better). +Figure 16: Performance improvement with LeaFTL. +SEATS +AMark +TPCC +OLTP +CompF +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Normalized Perf. +DFTL +SFTL +LeaFTL +Figure 17: Performance on the real SSD prototype. +99.9% +99% +90% +60% +30% +0% +Percentage of Storage Accesses +100 +101 +102 +103 +Latency ( s) +DFTL +SFTL +LeaFTL +Figure 18: The latency distribution of storage accesses when +running OLTP workload on the real SSD prototype. +segments are 100% accurate). This is because LeaFTL captures more +LPA-PPA mapping patterns. +We now evaluate the performance benefit of LeaFTL from its +saved memory space. We evaluate LeaFTL with two experimental +settings: (1) the SSD DRAM is mainly used (as much as possible) +for the mapping table; (2) the SSD DRAM is partially used for the +mapping table, in which we ensure at least 20% of the DRAM will +be used for the data caching. +In the first setting, DRAM is almost used for mapping table in +DFTL. As shown in Figure 16 (a), LeaFTL reduces the storage access +latency by 1.6× on average (up to 2.7×), compared to SFTL. This +is because LeaFTL saves more memory from the mapping table + +LeaFTL: A Learning-based Flash-Translation Layer for Solid-State Drives +MSR-hm +MSR-src2 +MSR-prxy +MSR-prn +MSR-usr +FIU-home +FIU-mail +SEATS +AMark +TPCC +OLTP +CompF +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Memory Footprint +Reduction +=0 +=1 +=4 +=16 +SSD Simulator +Real SSD +Figure 19: The reduction of the mapping table size of LeaFTL +with different 𝛾 (lower is better). +=0 +=1 +=4 +=16 +0% +20% +40% +60% +80% +100% +Percentage of +Segments +Accurate +Approximate +Figure 20: The distribution of learned segments. +than SFTL. SFTL slightly outperforms DFTL, because it reduces the +mapping table size by compressing mapping entries with grouping +strictly sequential data accesses. In the second setting, as shown in +Figure 16 (b), LeaFTL obtains 1.4× (up to 3.4×) and 1.6× (up to 4.9×) +performance speedup, compared to SFTL and DFTL, respectively. +4.3 +Benefits on the Real SSD Prototype +We validate the benefits of LeaFTL on the real SSD prototype with +real workloads (see Table 2). They include filesystem benchmark +suite FileBench [59], and transactional database workloads from +BenchBase [13, 61]. All these workloads run on the ext4 file system. +With FileBench, we run OLTP and CompFlow (CompF) workloads +to read/write 10GB files. With BenchBase, we run TPCC, Auction- +Mark (AMark), and SEATS workloads on MySQL, and their data- +base sizes are 10–30GB. These database workloads will generate +37–230GB read traffic and 26–59GB write traffic to the SSD. We allo- +cate 256MB DRAM to host the mapping table (for different DRAM +sizes, see our sensitivity analysis in §4.4). +We present the performance benefit of LeaFTL in Figure 17. +Across all workloads, LeaFTL obtains 1.4× performance speedup +on average (up to 1.5×), compared to SFTL and DFTL. Similar to +our evaluation with the SSD simulator implementation, the per- +formance benefit of LeaFTL comes from the memory saving from +the address mapping table. And LeaFTL demonstrates comparable +performance improvement on real SSD devices, in comparison with +the SSD simulator in §4.2. We also show the latency distribution of +storage accesses in Figure 18, when running the OLTP workload on +the real SSD prototype. In comparison with existing FTL schemes, +LeaFTL does not increase the tail latency of storage accesses. And +the higher cache hit ratio of LeaFTL brings latency reduction for +many storage accesses. +4.4 +Sensitivity Analysis +Vary the value of 𝛾. As we increase the value of 𝛾 from 0 to +16, the size of the learned mapping table is reduced, as shown in +MSR-hm +MSR-src2 +MSR-prxy +MSR-prn +MSR-usr +FIU-home +FIU-mail +SEATS +AMark +TPCC +OLTP +CompF +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Normalized Perf. +=0 +=1 +=4 +=16 +SSD Simulator +Real SSD +Figure 21: Performance with various 𝛾 (lower is better). +256MB +512MB +1024MB +(a) Various DRAM size +0.0 +0.5 +1.0 +Normalized Perf. +4KB +8KB +16KB +(b) Various flash page size +0.0 +0.5 +1.0 +Normalized Perf. +DFTL +SFTL +LeaFTL +Figure 22: SSD performance with different DRAM capacity +and flash page size (lower is better). +Figure 19. LeaFTL achieves 1.3× reduction on average (1.2× on +the real SSD) with 𝛾 = 16, compared to that of 𝛾 = 0. The saved +memory with a larger 𝛾 is achieved by learning a wider range +of LPAs into approximate segments. To further understand this, +we profile the distribution of segments learned by LeaFTL with +different values of 𝛾, as shown in Figure 20. When 𝛾 = 0, all the +segments are accurate. When 𝛾 = 16, 26.5% of the learned segments +are approximate on average, and LeaFTL delivers 1.3× improvement +on storage performance (1.2× with workloads on the real SSD), in +comparison with the case of 𝛾 = 0 (see Figure 21). +Vary the SSD DRAM capacity. We now conduct the sensitivity +analysis of SSD DRAM by varying its capacity from 256MB to 1GB +on the real SSD prototype. As shown in Figure 22 (a), LeaFTL always +outperforms DFTL and SFTL as we vary the SSD DRAM capacity. +As we increase the DRAM capacity, the storage workloads are still +bottlenecked by the available memory space for the data caching. +LeaFTL can learn various data access patterns and significantly +reduce the address mapping table size, the saved memory further +benefits data caching. +Vary the flash page size. In this experiment, we fix the number +of flash pages, and vary the flash page size from 4KB to 16KB in the +SSD simulator, as SSD vendors usually use larger flash pages for +increased SSD capacity. We use the simulator for this study, since +the flash page size of the real SSD is fixed. As shown in Figure 22 +(b), LeaFTL always performs the best in comparison with DFTL and +SFTL. As we increase the flash page size to 16KB, we can cache less +number of flash pages with limited DRAM capacity. Thus, LeaFTL +experiences a slight performance drop. As we fix the total SSD + +Jinghan Sun, Shaobo Li, Yunxin Sun, Chao Sun, Dejan Vucinic, and Jian Huang +1 +5 +10 +15 +20 +25 +30 +35 +(a) Number of Levels +99.99% +99.9% +99% +90% +0% +Percentage of +Lookups +MSR-prn +MSR-usr +MSR-src2 +MSR-hm +MSR-prxy +FIU-home +FIU-mail +0.0 +0.5 +1.0 +1.5 +(b) LPA Lookup Overhead (%) +99.99% +99.9% +99% +90% +0% +Percentage of +Lookups +SEATS +CompF +OLTP +TPCC +AMark +Figure 23: Performance overhead of the LPA lookup. +MSR-hm +MSR-src2 +MSR-prxy +MSR-prn +MSR-usr +FIU-home +FIU-mail +SEATS +AMark +TPCC +OLTP +CompF +0 +5 +10 +15 +20 +Misprediction (%) +=0 +=1 +=4 +=16 +SSD Simulator +Real SSD +Figure 24: Misprediction ratio of flash pages access. +capacity and vary the page size, LeaFTL outperforms SFTL by 1.2× +and 1.1× for the page size of 8KB and 16KB, respectively. +4.5 +Overhead Source in LeaFTL +We evaluate the overhead sources in LeaFTL in three aspects: (1) +the performance overhead of the learning procedure in LeaFTL; +(2) the LPA lookup overhead in the learned segments; and (3) the +overhead caused by the address misprediction in LeaFTL. +We evaluate the performance of segment learning and address +lookup on an ARM Cortex-A72 core. This core is similar to the +storage processor used in modern SSDs. The learning time for a +batch of 256 mapping entries is 9.8–10.8 𝜇s (see Table 3). As we +learn one batch of index segments for every 256 flash writes, the +learning overhead is only 0.02% of their flash write latency. +In LeaFTL, the LPA lookup is 40.2–67.5 ns, as the binary search of +segments is fast and some segments can be cached in the processor +cache. The lookup time is slightly higher as we increase𝛾, due to the +additional CRB accesses. We also profile the cumulative distribution +function (CDF) of the number of levels to lookup for each LPA +lookup, and present the results in Figure 23 (a). For most of the +tested workloads, 90% of the mapping table lookup can be fulfilled +at the topmost level, and 99% of the lookups are within 10 levels. +Although MSR-prn workload requires more lookups than other +workloads, it only checks 1.4 levels on average. We also evaluate +the performance overhead of the LPA lookup on the real SSD, and +show the results in Figure 23 (b). The extra lookup overhead for each +flash read is 0.21% on average. And for 99.99% of all the lookups, +the additional overhead is less than 1% of the flash access latency. +Table 3: Overhead source of LeaFTL with an ARM core. +𝛾 +0 +1 +4 +Learning (256 LPAs) +9.8 𝜇s +10.8 𝜇s +10.8 𝜇s +Lookup (per LPA) +40.2 ns +60.5 ns +67.5 ns +LeaFTL also has low misprediction ratios with approximate seg- +ments. This is because LeaFTL can still learn accurate segments +even if 𝛾 > 0, and not all entries in the approximate segments +will result in misprediction. As shown in Figure 24, most of the +workloads achieve less than 10% misprediction ratio when 𝛾 = 16. +We obtain similar misprediction ratio on the real SSD prototype. +Note that each misprediction only incurs one flash read access with +the help of our proposed OOB verification. +4.6 +Impact on SSD Lifetime +The flash blocks of an SSD can only undergo a certain amount of +writes. In this experiment, we use the write amplification factor +(WAF, the ratio between the actual and requested flash writes) to +evaluate the SSD lifetime. The SSD will age faster if the WAF is +larger. As shown Figure 25, the WAF of LeaFTL is comparable to +DFTL and SFTL. DFTL has larger WAF in most workloads. SFTL +and LeaFTL occasionally flush translation pages to the flash blocks, +but the cost is negligible. +5 +DISCUSSION +Why Linear Regression. Unlike deep neural networks, the lin- +ear regression used in LeaFTL is simple and lightweight, which +takes only a few microseconds to learn an index segment with +embedded ARM processors available in modern SSD controllers. +In addition, the linear regression algorithm has been well studied, +and offers guaranteed error bounds for its learned results. LeaFTL +is the first work that uses learning techniques to solve a critical +system problem (i.e., address mapping) in SSDs. +Adaptivity of LeaFTL. LeaFTL focuses on the page-level address +translation, its design and implementation will not be affected by +the low-level flash memory organization (i.e., TLC/QLC). As we +use TLC/QLC technique to further increase the SSD capacity, the +address mapping issue will become more critical, since the SSD +DRAM capacity does not scale well and becomes the bottleneck for +caching address mappings and user data. +Recovery of Learned Index Segments. As discussed in §3.8, us- +ing a battery or large capacitor to preserve and persist the cached +segments upon failures or crashes will simplify the recovery pro- +cedure significantly. In our real SSD prototype, we do not assume +the battery-backed DRAM is available. Thus, we follow the conven- +tional recovery approach in modern SSDs [20, 23], and scan flash +blocks in parallel by utilizing the channel-level parallelism. +When we run real workloads like TPCC on the SSD prototype, +we intentionally reboot the system after running the workload for +a period of time (0.5-3 hours). We find that the system can recover +in 15.8 minutes on average whenever the reboot happens. This +is similar to the time of recovering the conventional page-level +mapping table in DFTL [20]. This is mostly caused by scanning the +blocks in a channel (70MB/s per channel in our SSD prototype), +and the time for reconstructing recently learned segments is rela- +tively low (101.3 milliseconds on average). We believe the recovery + +LeaFTL: A Learning-based Flash-Translation Layer for Solid-State Drives +MSR-hm +MSR-src2 +MSR-prxy +MSR-prn +MSR-usr +FIU-home +FIU-mail +SEATS +AMark +TPCC +OLTP +CompF +0.0 +0.5 +1.0 +1.5 +Write +Amplification +DFTL +SFTL +LeaFTL +SSD Simulator +Real SSD +Figure 25: Write amplification factor of LeaFTL. +time is not much of a concern as the recovery does not happen +frequently in reality. And the recovery can be accelerated as we +increase the channel-level bandwidth. In addition, if an SSD can +tolerate more data losses, we can still ensure the crash consistency +by only loading the stored index segments from flash chips, which +requires minimum recovery time. +6 +RELATED WORK +Address Translation for SSDs. A variety of FTL optimizations +have been proposed [8, 12, 20, 25, 28, 34, 49, 50]. These works ex- +ploited the data locality of flash accesses to improve the cache +efficiency of the mapping table. However, most of them were devel- +oped with human-driven heuristics. An alternative approach is to +integrate application semantics into the FTL, such as content-aware +FTL [7]. However, they were application specific and required signif- +icant changes to the FTL. LeaFTL is a generic solution and does not +require application semantics in its learning. Researchers proposed +to integrate the FTL mapping table into the host [18, 23, 26, 66]. Typi- +cal examples include DFS [26], Nameless writes [66], FlashMap [23], +and FlatFlash [4]. LeaFTL is orthogonal to them and can be applied +to further reduce their memory footprint. +Machine Learning for Storage. Recent studies have been using +learning techniques to build indexes such as B-trees, log-structured +merge tree, hashmaps, and bloom filters [11, 14, 15, 32, 33, 42] +for in-memory datasets, identify optimal cache replacement and +prefetching policies [40, 53, 56, 57], facilitate efficient storage har- +vesting [52], and drive the development of software-defined stor- +age [24]. LeaFTL applies learning techniques to optimize the address +mapping. However, unlike existing optimizations [43, 63] such as +learned page table for virtual memory that used deep neural net- +works to learn the patterns, LeaFTL provides a lightweight solution. +SSD Hardware Development. For the recent SSD innovations [3, +17, 19, 47] like Z-SSD [55], KVSSD [35], and ZNS SSD [21], DRAM +capacity and storage processor are still the main constraints in SSD +controllers. As we scale the storage capacity, the challenge with +the address translation becomes only worse. Researchers recently +deployed hardware accelerators inside SSD controllers for near- +data computing [36, 41, 54, 58]. We wish to extend LeaFTL with +in-storage accelerators to deploy more powerful learning models +as the future work. +7 +CONCLUSION +We present a learning-based flash translation layer, named LeaFTL +for SSDs. LeaFTL can automatically learn different flash access +patterns and build space-efficient indexes, which reduces the ad- +dress mapping size and improves the caching efficiency in the SSD +controller. Our evaluation shows that LeaFTL improves the SSD +performance by 1.4× on average for a variety of storage workloads. +ACKNOWLEDGMENTS +We thank the anonymous reviewers for their helpful comments +and feedback. This work is partially supported by the NSF CAREER +Award 2144796, CCF-1919044, and CNS-1850317. +REFERENCES +[1] 2019. A Closer Look At SSD Power Loss Protection. https://www.kingston.com/ +en/blog/servers-and-data-centers/ssd-power-loss-protection. +[2] 2020. Harnessing Microcontrollers to Deliver Intelligent SSD Power Management +and PLP Capabilities. https://www.atpinc.com/de/about/stories/microcontroller- +SSD-power-loss-protection. +[3] 3D NAND – An Overview. 2022. +https://www.simms.co.uk/tech-talk/3d-nand-overview/. +[4] Ahmed Abulila, Vikram Sharma Mailthoday, Zaid Qureshi, Jian Huang, Nam Sung +Kim, Jin jun Xiong, and Wen mei Hwu. 2019. FlatFlash: Exploiting the Byte- +Accessibility of SSDs within A Unified Memory-Storage Hierarchy. In Proceedings +of the 24th ACM International Conference on Architectural Support for Programming +Languages and Operating Systems (ASPLOS’19). Providence, RI. +[5] Nitin Agrawal, Vijayan Prabhakaran, Ted Wobber, John D. Davis, Mark Manasse, +and Rina Panigrahy. 2008. Design Tradeoffs for SSD Performance. In Proceedings +of the USENIX 2008 Annual Technical Conference (ATC’08). Boston, Massachusetts. +[6] Yu Cai, Saugata Ghose, Erich F Haratsch, Yixin Luo, and Onur Mutlu. 2017. Error +characterization, mitigation, and recovery in flash-memory-based solid-state +drives. Proc. IEEE 105, 9 (2017), 1666–1704. +[7] Feng Chen, Tian Luo, and Xiaodong Zhang. 2011. CAFTL: A Content-Aware +Flash Translation Layer Enhancing the Lifespan of Flash Memory based Solid +State Drives. In Proceedings of the 9th USENIX Conference on File and Storage +Technologies (FAST’11). San Jose, CA. +[8] Renhai Chen, Zhiwei Qin, Yi Wang, Duo Liu, Zili Shao, and Yong Guan. 2014. On- +demand block-level address mapping in large-scale NAND flash storage systems. +IEEE Trans. Comput. 64, 6 (2014), 1729–1741. +[9] Tae-Sun Chung, Dong-Joo Park, and Jongik Kim. 2011. LSTAFF*: An Efficient +Flash Translation Layer for Large Block Flash Memory. In Proceedings of the 2011 +ACM Symposium on Applied Computing (SAC’11). TaiChung Taiwan. +[10] Curtis R Cook and Do Jin Kim. 1980. Best sorting algorithm for nearly sorted +lists. Commun. ACM 23, 11 (1980), 620–624. +[11] Yifan Dai, Yien Xu, Aishwarya Ganesan, Ramnatthan Alagappan, Brian Kroth, +Andrea Arpaci-Dusseau, and Remzi Arpaci-Dusseau. 2020. From WiscKey to +Bourbon: A Learned Index for Log-Structured Merge Trees. In Proceedings of +the 14th USENIX Symposium on Operating Systems Design and Implementation +(OSDI’20). Virtual Event. +[12] Niv Dayan, Philippe Bonnet, and Stratos Idreos. 2016. GeckoFTL: Scalable Flash +Translation Techniques For Very Large Flash Devices. In Proceedings of the Inter- +national Conference on Management of Data (SIGMOD’16). San Francisco, CA. +[13] Djellel Eddine Difallah, Andrew Pavlo, Carlo Curino, and Philippe Cudré- +Mauroux. 2013. OLTP-Bench: An Extensible Testbed for Benchmarking Relational +Databases. PVLDB 7, 4 (2013). +[14] Paolo Ferragina, Fabrizio Lillo, and Giorgio Vinciguerra. 2020. Why Are Learned +Indexes So Effective?. In Proceedings of the 37th International Conference on +Machine Learning (ICML’20). Virtual Event. +[15] Paolo Ferragina and Giorgio Vinciguerra. 2020. The PGM-Index: A Fully-Dynamic +Compressed Learned Index with Provable Worst-Case Bounds. Proceedings of +the VLDB Endowment 13, 8 (April 2020). +[16] FIU. 2009. FIU Server Traces. +[17] Flash Memory. 2022. https://en.wikipedia.org/wiki/Flash_memory. +[18] Fusion-io Directcache: Transparent Storage Accelerator. 2011. +http://www.fusionio.com/systems/directcache/. +[19] Gartner. 2017. Forecast Overview: NAND Flash, Worldwide, 2017. +https: +//www.gartner.com/doc/3745121/forecast-overview-nand-flash-worldwide +[20] Aayush Gupta, Youngjae Kim, and Bhuvan Urgaonkar. 2009. DFTL: A Flash +Translation Layer Employing Demand-based Selective Caching of Page-level +Address Mappings. In Proceedings of the 14th International Conference on Archi- +tectural Support for Programming Languages and Operating Systems (ASPLOS’09). +Washington, DC. +[21] Kyuhwa Han, Hyunho Gwak, Dongkun Shin, and Joo-Young Hwang. 2021. ZNS+: +Advanced Zoned Namespace Interface for Supporting In-Storage Zone Com- +paction. In 15th {USENIX} Symposium on Operating Systems Design and Imple- +mentation (OSDI’21). 147–162. +[22] Jian Huang, Anirudh Badam, Laura Caulfield, Suman Nath, Sudipta Sengupta, +Bikash Sharma, and Moinuddin K. Qureshi. 2017. FlashBlox: Achieving Both +Performance Isolation and Uniform Lifetime for Virtualized SSDs. In Proceedings + +Jinghan Sun, Shaobo Li, Yunxin Sun, Chao Sun, Dejan Vucinic, and Jian Huang +of the 15th Usenix Conference on File and Storage Technologies (FAST’17). Santa +clara, CA. +[23] Jian Huang, Anirudh Badam, Moinuddin K. Qureshi, and Karsten Schwan. 2015. +Unified Address Translation for Memory-mapped SSDs with FlashMap. In Pro- +ceedings of the 42nd Annual International Symposium on Computer Architecture +(ISCA’15). Portland, OR. +[24] Jian Huang, Daixuan Li, and Jinghan Sun. 2022. Learning to Drive Software- +Defined Storage. Workshop on Machine Learning for Systems at NIPS’22 (2022). +[25] Song Jiang, Lei Zhang, XinHao Yuan, Hao Hu, and Yu Chen. 2011. S-FTL: An +Efficient Address Translation for Flash Memory by Exploiting Spatial Locality. +In Proceedings of the 2011 IEEE 27th Symposium on Mass Storage Systems and +Technologies (MSST’11). IEEE Computer Society. +[26] William K. Josephson, Lars A. Bongo, Kai Li, and David Flynn. 2010. DFS: A +File System for Virtualized Flash Storage. ACM Trans. on Storage 6, 3 (2010), +14:1–14:25. +[27] Jun He, Sudarsun Kannan, Andrea C. Arpaci-Dusseau, Remzi H. Arpaci-Dusseau. +2017. The Unwritten Contract of Solid State Drives. In Proceedings of the Twelfth +European Conference on Computer Systems (EuroSys’17). Belgrade, Serbia. +[28] Dawoon Jung, Jeong-UK Kang, Heeseung Jo, Jin-Soo Kim, and Joonwon Lee. +2010. Superblock FTL: A superblock-based flash translation layer with a hybrid +address translation scheme. ACM Transactions on Embedded Computing Systems +(TECS) 9, 4 (2010), 1–41. +[29] Jeong-Uk Kang, Heeseung Jo, Jinsoo Kim, and Joonwon Lee. 2006. A Superblock- +Based Flash Translation Layer for NAND Flash Memory. In Proceedings of the +6th International Conference on Embedded Software (EMSOFT’06). Seoul, South +Korea. +[30] Luyi Kang, Yuqi Xie, Weiwei Jia, Xiaohao Wang, Jongryool Kim, Changhwan +Youn, Myeong Joon Kang, Jin Lim, Bruce Jacob, and Jian Huang. 2021. IceClave: A +Trusted Execution Environment for In-Storage Computing. In Proceedings of the +54th Annual IEEE/ACM International Symposium on Microarchitecture (MICRO’21). +Virtual Event. +[31] Jesung Kim, Jong Min Kim, S.H. Noh, Sang Lyul Min, and Yookun Cho. 2002. A +space-efficient flash translation layer for CompactFlash systems. IEEE Transac- +tions on Consumer Electronics 48, 2 (2002). +[32] Andreas Kipf, Ryan Marcus, Alexander van Renen, Mihail Stoian, Alfons Kemper, +Tim Kraska, and Thomas Neumann. 2020. RadixSpline: A Single-Pass Learned +Index. In Proceedings of the Third International Workshop on Exploiting Artificial +Intelligence Techniques for Data Management (aiDM ’20). Portland, Oregon. +[33] Tim Kraska, Alex Beutel, Ed H. Chi, Jeffrey Dean, and Neoklis Polyzotis. 2018. +The Case for Learned Index Structures. In Proceedings of the 2018 International +Conference on Management of Data (SIGMOD’18). Houston, TX, USA. +[34] Hunki Kwon, Eunsam Kim, Jongmoo Choi, Donghee Lee, and Sam H Noh. 2010. +Janus-FTL: Finding the optimal point on the spectrum between page and block +mapping schemes. In Proceedings of the tenth ACM international conference on +Embedded software. 169–178. +[35] Samsung Memory Solutions Lab. 2017. Samsung Key Value SSD enables High Per- +formance Scaling. https://www.samsung.com/semiconductor/global.semi.static/ +Samsung_Key_Value_SSD_enables_High_Performance_Scaling-0.pdf (2017). +[36] Joo Hwan Lee, Hui Zhang, Veronica Lagrange, Praveen Krishnamoorthy, Xi- +aodong Zhao, and Yang Seok Ki. 2020. SmartSSD: FPGA accelerated near-storage +data analytics on SSD. IEEE Computer architecture letters 19, 2 (2020), 110–113. +[37] Sungjin Lee, Ming Liu, Sangwoo Jun, Shuotao Xu, Jihong Kim, and Arvind. 2016. +Application-managed flash. In Proceedings of the 14th USENIX Conference on File +and Storage Technologies (FAST’16). 339–353. +[38] Sungjin Lee, Dongkun Shin, Young-Jin Kim, and Jihong Kim. 2008. LAST: Locality- +Aware Sector Translation for NAND Flash Memory-Based Storage Systems. In +Proceedings of the SIGOPS Operating Systems Review (2008). +[39] Sang-Won Lee, Dong-Joo Park, Tae-Sun Chung, Dong-Ho Lee, Sangwon Park, +and Ha-Joo Song. 2007. A Log Buffer-Based Flash Translation Layer Using +Fully-Associative Sector Translation. ACM Transactions on Embedded Computing +Systems 6, 3 (2007), 18:1–18:27. +[40] Evan Liu, Milad Hashemi, Kevin Swersky, Parthasarathy Ranganathan, and Jun- +whan Ahn. 2020. An imitation learning approach for cache replacement. In +International Conference on Machine Learning. PMLR, 6237–6247. +[41] Vikram Sharma Mailthoday, Zaid Qureshi, Weixin Liang, Ziyan Feng, Simon Gar- +cia de Gonzalo, Youjie Li, Hubertus Franke, Jinjun Xiong, Jian Huang, and Wen +mei Hwu. 2019. DeepStore: In-Storage Acceleration for Intelligent Queries. In +Proceedings of the 52nd IEEE/ACM International Symposium on Microarchitecture +(MICRO’19). Columbus, OH. +[42] Ryan Marcus, Emily Zhang, and Tim Kraska. 2020. CDFShop: Exploring and +Optimizing Learned Index Structures. In Proceedings of the 2020 ACM SIGMOD +International Conference on Management of Data (SIGMOD’20). Portland, OR, USA. +https://doi.org/10.1145/3318464.3384706 +[43] Artemiy Margaritov, Dmitri Ustiugov, Edouard Bugnion, and Boris Grot. 2018. +Virtual Address Translation via Learned Page Table Indexes. In Proceedings of +the Workshop on ML for Systems at NeurIPS. Montreal, Canada. +[44] Kiran Kumar Matam, Gunjae Koo, Haipeng Zha, Hung-Wei Tseng, and Murali +Annavaram. 2019. GraphSSD: Graph Semantics Aware SSD. In Proceedings of +the 46th International Symposium on Computer Architecture (ISCA’19). Phoenix, +Arizona. +[45] Microsoft. 2007. MSR Cambridge Traces. +[46] Jian Ouyang, Shiding Lin, Song Jiang, Yong Wang, Wei Qi, Jason Cong, and +Yuanzheng Wang. 2014. SDF: Software-Defined Flash for Web-Scale Internet +Storage Systems. In Proceedings of 19th International Conference on Architectural +Support for Programming Language and Operating Systems (ASPLOS’14). Salt Lake +City, UT. +[47] Over 50 years of development history of Flash Memory Technology. 2019. +https://www.elinfor.com/knowledge/over-50-years-of-development-history- +of-flash-memory-technology-p-11271. +[48] Nikolaos Papandreou, Haralampos Pozidis, Nikolas Ioannou, Thomas Parnell, +Roman Pletka, Milos Stanisavljevic, Radu Stoica, Sasa Tomic, Patrick Breen, Gary +Tressler, et al. 2020. Open block characterization and read voltage calibration of +3D QLC NAND flash. In 2020 IEEE International Reliability Physics Symposium +(IRPS). IEEE, 1–6. +[49] Chanik Park, Wonmoon Cheon, Jeonguk Kang, Kangho Roh, Wonhee Cho, and +Jin-Soo Kim. 2008. A reconfigurable FTL (flash translation layer) architecture +for NAND flash-based applications. ACM Transactions on Embedded Computing +Systems (TECS) 7, 4 (2008), 1–23. +[50] Zhiwei Qin, Yi Wang, Duo Liu, and Zili Shao. 2010. Demand-based block-level +address mapping in large-scale NAND flash storage systems. In Proceedings of +the eighth IEEE/ACM/IFIP international conference on Hardware/software codesign +and system synthesis. +[51] Benjamin Reidys, Peng Liu, and Jian Huang. 2022. RSSD: Defend against Ran- +somware with Hardware-Isolated Network-Storage Codesign and Post-Attack +Analysis. In Proceedings of the 27th ACM International Conference on Architec- +tural Support for Programming Languages and Operating Systems (ASPLOS’22). +Lausanne, Switzerland. +[52] Benjamin Reidys, Jinghan Sun, Anirudh Badam, Shadi Noghabi, and Jian Huang. +2022. BlockFlex: Enabling Storage Harvesting with Software-Defined Flash +in Modern Cloud Platforms. In Proceedings of the 16th USENIX Symposium on +Operating Systems Design and Implementation (OSDI’22). Carlsbad, CA. +[53] Liana V Rodriguez, Farzana Yusuf, Steven Lyons, Eysler Paz, Raju Rangaswami, +Jason Liu, Ming Zhao, and Giri Narasimhan. 2021. Learning Cache Replacement +with CACHEUS. In 19th USENIX Conference on File and Storage Technologies +(FAST’21). 341–354. +[54] Zhenyuan Ruan, Tong He, and Jason Cong. 2019. INSIDER: Designing In-Storage +Computing System for Emerging High-Performance Drive. In Proceedings of the +2019 USENIX Annual Technical Conference (USENIX ATC’19). Renton, WA. +[55] Samsung Z-NAND. 2019. https://www.samsung.com/semiconductor/ssd/z-ssd/. +[56] Subhash Sethumurugan, Jieming Yin, and John Sartori. 2021. Designing a Cost- +Effective Cache Replacement Policy using Machine Learning. In 2021 IEEE Inter- +national Symposium on High-Performance Computer Architecture (HPCA). IEEE, +291–303. +[57] Zhan Shi, Xiangru Huang, Akanksha Jain, and Calvin Lin. 2019. Applying deep +learning to the cache replacement problem. In Proceedings of the 52nd Annual +IEEE/ACM International Symposium on Microarchitecture. 413–425. +[58] smartssd 2018. SmartSSD Computational Storage Drive. https://www.xilinx.com/ +applications/data-center/computational-storage/smartssd.html. +[59] Vasily Tarasov, Erez Zadok, and Spencer Shepler. 2016. Filebench: A flexible +framework for file system benchmarking. The USENIX Magazine 41, 1 (2016). +[60] Usman Saleem, Advanced SSD Buying Guide - NAND Types, DRAM Cache, HMB +Explained. 2022. https://appuals.com/ssd-buying-guide/. +[61] Dana Van Aken, Djellel E. Difallah, Andrew Pavlo, Carlo Curino, and Philippe +Cudré-Mauroux. 2015. BenchPress: Dynamic Workload Control in the OLTP- +Bench Testbed. In Proceedings of the 2015 ACM SIGMOD International Conference +on Management of Data (SIGMOD’15). +[62] Xiaohao Wang, Yifan Yuan, You Zhou, Chance C. Coats, and Jian Huang. 2019. +Project Almanac: A Time-Traveling Solid-State Drive. In Proceedings of the 14th +European Conference on Computer Systems (EuroSys’19). Dresden, Germany. +[63] Nan Wu and Yuan Xie. 2021. A Survey of Machine Learning for Computer +Architecture and Systems. CoRR abs/2102.07952 (2021). https://arxiv.org/abs/ +2102.07952 +[64] Qing Xie, Chaoyi Pang, Xiaofang Zhou, Xiangliang Zhang, and Ke Deng. 2014. +Maximum Error-Bounded Piecewise Linear Representation for Online Stream +Approximation. Proceedings of the VLDB Journal 23, 6 (Dec. 2014). +[65] Qing Xie, Chaoyi Pang, Xiaofang Zhou, Xiangliang Zhang, and Ke Deng. 2014. +Maximum error-bounded piecewise linear representation for online stream ap- +proximation. The VLDB journal 23, 6 (2014), 915–937. +[66] Yiying Zhang, Leo Prasath Arulraj, Andrea C. Arpaci-Dusseau, and Remzi H. +Arpaci-Dusseau. 2012. De-indirection for Flash-based SSDs with Nameless Writes. +In Proceedings of the 10th USENIX Conference on File and Storage Technologies +(FAST’12). San Jose, CA. + diff --git a/A9AyT4oBgHgl3EQfRvd3/content/tmp_files/load_file.txt b/A9AyT4oBgHgl3EQfRvd3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5c8d06f257447efc53651f6cb38ad38d00577c6e --- /dev/null +++ b/A9AyT4oBgHgl3EQfRvd3/content/tmp_files/load_file.txt @@ -0,0 +1,1207 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf,len=1206 +page_content='LeaFTL: A Learning-based Flash Translation Layer for Solid-State Drives Jinghan Sun UIUC js39@illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='edu Shaobo Li UIUC shaobol2@illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='edu Yunxin Sun∗ ETH Zurich yunsun@student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='ch Chao Sun Western Digital Research chao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='sun@wdc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='com Dejan Vucinic Western Digital Research dejan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='vucinic@wdc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='com Jian Huang UIUC jianh@illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='edu ABSTRACT In modern solid-state drives (SSDs), the indexing of flash pages is a critical component in their storage controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' It not only affects the data access performance, but also determines the efficiency of the precious in-device DRAM resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' A variety of address mapping schemes and optimizations have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' However, most of them were developed with human-driven heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In this paper, we present a learning-based flash translation layer (FTL), named LeaFTL, which learns the address mapping to tolerate dynamic data access patterns via linear regression at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' By grouping a large set of mapping entries into a learned segment, it significantly reduces the memory footprint of the address mapping table, which further benefits the data caching in SSD controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL also employs various optimization techniques, including out-of-band metadata verification to tolerate mispredictions, opti- mized flash allocation, and dynamic compaction of learned index segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We implement LeaFTL with both a validated SSD sim- ulator and a real open-channel SSD board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Our evaluation with various storage workloads demonstrates that LeaFTL saves the memory consumption of the mapping table by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='9× and improves the storage performance by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='4× on average, in comparison with state-of-the-art FTL schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' CCS CONCEPTS Hardware → External storage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' • Computer systems orga- nization → Architectures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' • Computing methodologies → Learning linear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' KEYWORDS Learning-Based Storage, Flash Translation Layer, Solid-State Drive 1 INTRODUCTION Flash-based SSDs have become an indispensable part in modern storage systems, as they outperform conventional hard-disk drives (HDDs) by orders of magnitude, and their cost is close to that of HDDs [22, 30, 51, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The SSD capacity continues to boost by increasing the number of flash channels and chips with the rapidly shrinking process and manufacturing technology [22, 25, 41, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The flash translation layer (FTL) is the core component of man- aging flash memory in SSDs, including address translation, garbage collection (GC), and wear leveling [20, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The FTL maintains meta- data structures for different functions such as address translation ∗Work done when visiting the Systems Platform Research Group at UIUC as a research intern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' and valid page tracking, and caches them in the in-device DRAM (SSD DRAM) for improved performance [7, 12, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Among these data structures, the address mapping table has the largest memory footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In general, the address mapping table can be categorized in three types: page-level mapping, block- level mapping, and hybrid mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Modern SSDs usually use the page-level mapping, as it offers the best performance for the flash page lookup, and incurs minimal GC overhead, in comparison with the other two mapping schemes [20, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' However, the page-level mapping table size is large, as it stores the entry for the LPA-to-PPA address translation for each flash page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The address mapping table significantly affects the performance of SSDs, as it not only determines the efficiency of indexing flash pages, but also affects the utilization of SSD DRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Moreover, due to the limitations of the cost and power budget in SSD controllers, it is challenging for SSD vendors to scale the in-device DRAM capacity [12, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' This challenge becomes even worse with the increasing flash memory capacity in an SSD, as larger capacity usually requires a larger address mapping table for indexing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' To improve the address mapping and translation for SSDs, vari- ous optimization schemes have been developed [9, 25, 29, 38, 39, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' However, most of them were developed based on human-driven heuristics [25], and cannot capture dynamic data access patterns at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Employing more semantic knowledge into the FTL, such as GraphSSD [44], can improve the data indexing and address translation, however, it is application specific and complicates the management of address mappings [7], which does not scale for the development of generic SSDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In this work, we do not expect that we can obtain application semantics from the host and the SSD con- troller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Instead, we focus on utilizing simple yet effective machine learning (ML) techniques to automate the address mapping table management in the SSDs, with the capability of learning diverse and dynamic data access patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' To this end, we propose a learning-based FTL, named LeaFTL, by utilizing the piecewise linear regression technique to learn the LPA- PPA mappings, and automatically exploiting the data locality of various data access patterns at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Unlike the state-of-the-art page-level mapping, the key idea of LeaFTL is that it can learn the correlation between a set of LPAs and their mapped PPAs, based on which it can build a space-efficient index segment, as presented in A in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Since the learned index segment can be simply represented with (𝑆, 𝐿, 𝐾, 𝐼), where [𝑆,𝑆 + 𝐿] denotes the interval of LPAs, 𝐾 is the slope of the segment, and 𝐼 is the intercept of the segment (see the last diagram in Figure 1), each segment will take arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='00072v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='OS] 30 Dec 2022 Jinghan Sun, Shaobo Li, Yunxin Sun, Chao Sun, Dejan Vucinic, and Jian Huang 30 LPA PPA 31 32 33 34 155 156 157 158 159 60 62 64 66 68 200 201 203 204 205 80 82 83 84 87 304 305 306 307 308 Index Segment A Index Segment B Index Segment C LPA PPA A B C error bound 1 1 1 1 2 2 2 2 2 1 1 3 Figure 1: An illustrative example of learning LPA-PPA mappings using piecewise linear regression in LeaFTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' It can learn various patterns of LPA-PPA mappings with guaranteed error bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Each learned index segment can be represented with (𝑆, 𝐿, 𝐾, 𝐼), where [𝑆,𝑆 + 𝐿] denotes the interval of LPAs, 𝐾 is the slope, and 𝐼 is the intercept of the index segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' only 8 bytes (1 byte for 𝑆 and 𝐿, 2 bytes for 𝐾, and 4 bytes for 𝐼) with our optimizations (see the details in §3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Compared to the on- demand page-level mapping [20], the learned segment reduces the mapping table size by a factor of 𝑚 ∗ 𝑎𝑣𝑔(𝐿)/8, where 𝑚 is the size (8 bytes) of each entry in the on-demand page-level mapping table, and 𝑎𝑣𝑔(𝐿) is the average number of LPA-PPA mappings that can be represented in a learned index segment, 𝑎𝑣𝑔(𝐿) is 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='3 according to our study of various storage workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Beyond learning contiguous LPA-PPA mappings, LeaFTL also learns different correlation patterns, such as regular and irregular strided data accesses as shown in B and C , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Unlike existing indexing optimizations based on human-driven heuristics, LeaFTL can learn more irregular patterns of LPA-PPA mappings with guaranteed error bound, as shown in C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' This enables LeaFTL to further condense the address mapping table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Therefore, given a limited DRAM capacity in the SSD controller, LeaFTL can maximally utilize the DRAM caching and improve the storage performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' For the worst case like random I/O accesses, LeaFTL will transfer the mapping into single-point linear segments (𝐿 = 0, 𝐾 = 0, and 𝐼 = 𝑃𝑃𝐴 in Figure 1), and its memory consumption will be no more than that of the page-level mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' With the learned index segments, LeaFTL may occasionally re- turn an inaccurate PPA (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=', address misprediction), which incurs additional flash accesses until the correct PPA is identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' To over- come this challenge, we develop an error-tolerant mechanism in LeaFTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' For each flash page access, we use the reverse mapping stored in the out-of-band (OOB) metadata of each flash page to verify the correctness of the data access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Since the OOB usually has 64–256 bytes [20, 23], we use it to store the accurate LPAs mapped to the neighbor PPAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Thus, upon an address misprediction, we use the stored reverse mappings to find the correct PPA, avoiding addi- tional flash accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL leverages the intrinsic OOB structure to handle address mispredictions and make SSD perfectly-suited for practical learned indexing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Due to the intrinsic out-of-place write property of SSDs (see §2), the learned index segments will be disrupted by writes and GC, and the segments need to be relearned with new LPA-PPA mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' To tolerate these disruptions, the learned segments are organized within multiple levels to maintain the temporal order in a log-structured manner: the topmost level has the most recent segments, and the lower level stores older segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The segments at the same level are sorted without overlapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' If the new segment has a conflict with an existing segment, the old segment will be moved to the lower level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Therefore, LeaFTL can always identify the latest version of the corresponding LPA-PPA mapping in a top level of learned index segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL will compact the learned segments periodically to reduce its memory footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' To further maximize the efficiency of LeaFTL, we coordinate its learning procedure with flash block allocation in the SSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As flash block allocation decides the distribution of mapped PPAs, LeaFTL will allocate consecutive PPAs to contiguous LPAs at its best effort, for increasing the possibility of learning a space-efficient index seg- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Similar to existing page-level mapping [20, 23], LeaFTL stores the learned index segments in flash blocks for recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Overall, we make the following contributions: We present a learning-based FTL, it can learn various data access patterns and turn them into index segments for reducing the storage cost of the mapping table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We develop an error-tolerant address translation mechanism to handle address mispredictions caused by the learned indexes, with minimal extra flash accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We preserve the core FTL functions, and enable the coordination between the learning procedure of the address mapping table with the flash block allocation and GC to maximize the efficiency of the learned FTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We manage the learned segments in an optimized log-structured manner, and enable compaction to further improve the space efficiency for the address mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We implement LeaFTL with a validated SSD simulator Wisc- Sim [27] and evaluate its efficiency with a variety of popular storage workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We also develop a system prototype with a real 1TB open-channel SSD to verify the functions of LeaFTL and validate its efficiency with real data-intensive applications, such as the key- value store and transactional database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Our evaluation with the real SSD shows similar benefits as that of the SSD simulator imple- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We demonstrate that LeaFTL reduces the storage cost of the address mapping in the FTL by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='9× on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The saved memory space benefits the utilization of the precious SSD DRAM, and further improves the storage performance by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='4× on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We also show that LeaFTL does not affect the SSD lifetime, and its LeaFTL: A Learning-based Flash-Translation Layer for Solid-State Drives flash flash flash flash Flash Flash Flash Flash DRAM Flash Controller SSD Controller/Firmware PCIe Interface Embedded Processor Internal Bus DRAM Controller Block I/O Figure 2: The internal system architecture of SSDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' learning procedure introduces negligible performance overhead to the storage processor in the SSD controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The codebase of LeaFTL is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='com/platformxlab/LeaFTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2 BACKGROUND AND MOTIVATION Flash-Based Solid-State Drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' An SSD has three major parts (see Figure 2): a set of flash memory packages, an SSD controller with embedded processors, and a set of flash controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' With the nature of NAND Flash, when a free page is written, the page cannot be written again until that page is erased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' However, erase operation is performed only at a block granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As the erase operation is expensive, writes are issued to free flash pages erased in advance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=', out-of-place write).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' GC will be performed to clean the stale data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As each flash block has limited endurance, it is important for them to age uniformly (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=', wear leveling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' SSDs have a logical- to-physical address mapping table to index flash pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' All these functions are managed by the FTL in the SSD firmware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Modern SSD controllers have general-purpose embedded pro- cessors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=', ARM processors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The processors help with issuing I/O requests, translating LPAs to PPAs, and handling GC and wear- leveling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' SSDs also have limited DRAM capacities to cache the mapping table and the application data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Address Mapping Table in the FTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The address mapping table in FTL generally has three types: page-level mapping, block-level mapping, and hybrid mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The page-level mapping enables di- rect LPA-PPA mapping for fast lookup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' However, each entry usually takes 8 bytes (4 bytes for LPA, 4 bytes for PPA), and the entire map- ping table requires large storage space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The block-level mapping significantly reduces the mapping table size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' However, it introduces additional overhead for the page lookup in the flash block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The hy- brid mapping takes advantages of both page-level and block-level mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' It uses log blocks to store new writes, and index them with the page-level mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The log blocks will be moved into data blocks that are indexed with block-level mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' This incurs significant GC overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Therefore, modern SSDs commonly use the page-level mapping scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Metadata Structures for Flash Management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The FTL usually employs four metadata structures (see Figure 3): (1) the address mapping cache ( 1 AMC) for caching the address mapping table in the SSD DRAM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' (2) the global mapping directory ( 2 GMD) for tracking the locations of the address mapping table pages in the Address Mapping Cache (AMC) 1 Global Mapping Directory (GMD) 2 Block Validity Counter (BVC) 3 Page Validity Table (PVT) 4 LPA PPA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LX PY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LPA PPA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' VX PZ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' PBA Counter .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' PBA Bitmap .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' PB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Data Structures in the FTL of Modern SSDs Flash Memory Data Blocks Address Mapping Blocks Validity Blocks Figure 3: The common data structures in the FTL of SSDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' SSD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' (3) the block validity counter ( 3 BVC) for tracking the number of valid pages for each flash block for assisting the GC in the SSD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' and (4) the page validity table ( 4 PVT), which uses bitmaps to track the valid pages in each flash block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' During the GC, the FTL will check the 3 BVC to select candidate flash blocks, and migrate their valid pages to free flash blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' After that, it will erase these selected flash blocks, and mark them as free blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Limited DRAM Capacity in SSD Controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' It is hard to provi- sion large DRAM inside SSD controllers, due to their hardware con- straints and limited budgets for power and hardware cost [12, 41, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Thus, SSD controllers often use on-demand caching to maintain the recently accessed metadata and data in the SSD DRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Among all the metadata structures, the address mapping table has the largest memory footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As discussed, 1 AMC caches the recently accessed mapping table entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' If a mapping entry is not cached, the FTL will locate the corresponding address mapping ta- ble pages stored in the flash blocks, and place the mapping entry in the 1 AMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As we scale the SSD capacity, the DRAM challenge will become even worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' To overcome this challenge, various optimiza- tions on the mapping table have been proposed [9, 25, 29, 31, 38, 39] to improve the utilization of the SSD DRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' However, most of them cannot automatically capture diverse data access patterns at runtime, leaving a large room for improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 3 DESIGN AND IMPLEMENTATION To develop LeaFTL in the SSD controller, we have to overcome the following research challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL should be able to automatically capture diverse data access patterns, and generate memory-efficient address mapping (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='1, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='2, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='3, and §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL may incur address mispredictions, which could incur additional flash accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL should be tolerant of errors and have low misprediction penalty (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL should work coordinately with other core FTL functions that include GC and wear leveling (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL should be lightweight and not incur much extra overhead to storage operations (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='7, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='8 and §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Jinghan Sun, Shaobo Li, Yunxin Sun, Chao Sun, Dejan Vucinic, and Jian Huang (a) Precise Linear Approximation (b) Inaccurate Linear Approximation Figure 4: Visualization of learned index segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 1 2 4 8 16 32 64 128 256 512 1024 2048 Length of Learned Segments 0 20 40 60 80 100 Percentage of Segments (%) =0, #Segments=5540 =4, #Segments=4267 =8, #Segments=3718 Figure 5: Aggregated distribution of learned segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='1 Key Ideas of LeaFTL Instead of using the space-consuming one-to-one mapping in the page-level mapping, the key idea of LeaFTL is to exploit learning techniques to identify various LPA-PPA mapping patterns and build efficient learned address mapping entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Modern SSD controllers usually have a data buffer for grouping writes and write the large data chunk at once for exploiting the internal flash parallelisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL utilizes this data buffer to collect LPA-to-PPA mappings for learning index segments for free, and does not introduce extra data collection overhead (see the details in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As shown in Figure 4 (a), the PPA of an LPA can be obtained with the expression: 𝑃𝑃𝐴 = 𝑓 (𝐿𝑃𝐴) = ⌈𝐾 ∗ 𝐿𝑃𝐴 + 𝐼⌉, 𝐿𝑃𝐴 ∈ [𝑆𝐿𝑃𝐴,𝑆𝐿𝑃𝐴 + 𝐿], where [𝑆𝐿𝑃𝐴,𝑆𝐿𝑃𝐴 + 𝐿] denotes the interval (𝐿) of LPAs, 𝐾 is the slope, and 𝐼 is the intercept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As discussed in §1, each learned index segment can be represented in 8 bytes: 1 byte for 𝑆𝐿𝑃𝐴 and 𝐿, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2 bytes for 𝐾, and 4 bytes for 𝐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The size of 𝑆𝐿𝑃𝐴 is reduced from 4 bytes to 1 byte with our optimizations on the segment management (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We can relax the linear regression to capture more flash access patterns, which further reduces the learned address mapping table size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As shown in Figure 4 (b), the linear regression can learn a pattern with guaranteed error bound [−𝛾,𝛾].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As we increase 𝛾, we can cover more flash access patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We applied the relaxed linear regression with different 𝛾 values to a variety of storage workloads (see §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='1), our experimental results demonstrate that the number of learned index segments is gradually decreased, as we increase 𝛾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Figure 5 shows that 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='2–99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='2% of the learned index segments cover Segment SLPA L K I 1B 1B 2B 4B Type LPAs PPAs Index Segment Accurate [0, 1, 2, 3] [32, 33, 34, 35] Approximate [0, 1, 4, 5] [64, 65, 66, 67] 0 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='00 32 0 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='56 64 Figure 6: Types of learned segments in LeaFTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' up to 128 LPA-PPA mapping entries, demonstrating the potential advantages of the learning-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As for random access patterns, LeaFTL will transfer the learned segments into single-point segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' And these linear segments do not require more storage space than the page-level mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='2 Learned Index Segment Types of Learned Index Segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The mapping table of LeaFTL is built with learned index segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' It has two types of segments: accurate and approximate segments, as shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Both of them are learned with piecewise linear regression technique [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As for the accurate index segments, given an LPA, we can pre- cisely get the corresponding PPA with 𝑓 (𝐿𝑃𝐴) = ⌈𝐾 ∗ 𝐿𝑃𝐴 + 𝐼⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' For example, when the LPA is 2 in Figure 6, we can directly get the PPA value of 34 with ⌈1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='00 ∗ 2 + 32⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In this example, the learned segment has 𝐿 = 3 and it indexes 4 LPA-PPA mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' If 𝐿 = 0, the learned segment will become a single-point segment, the slope 𝐾 = 0, and we will get its PPA with 𝑃𝑃𝐴 = 𝐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As for approximate index segments, we use the same formula 𝑓 (𝐿𝑃𝐴) = ⌈𝐾 ∗𝐿𝑃𝐴+𝐼⌉ to calculate the PPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' However, the returned PPA may not be the exact corresponding PPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' It has an error bound [−𝛾,𝛾] guaranteed by the linear regression, and 𝛾 is configurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' For example, given 𝐿𝑃𝐴 = 4 in Figure 6, the value of the PPA is 67, according to the calculation ⌈4 ∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='56 + 64⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' However, the real PPA should be 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We define this as address misprediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We will discuss how we handle the address misprediction with reduced miss penalty in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Size of Learned Index Segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As discussed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='1, each seg- ment can be expressed in (𝑆𝐿𝑃𝐴, 𝐿, 𝐾, 𝐼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The starting LPA will take 4 bytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We can further reduce this size by partitioning a range of LPAs into small groups, and each LPA group represents a certain number of contiguous LPAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Therefore, we can index an LPA with its offset in a corresponding group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In LeaFTL, each group repre- sents 256 contiguous LPAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Thus, 𝑆𝐿𝑃𝐴 can be indexed by the offset (28 = 256) in the group, which takes only 1 byte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We use 256 as the group size, because the length of the learned segments is usually less than 256 (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Given an LPA, we can get its offset in the group with (𝐿𝑃𝐴 𝑚𝑜𝑑 256).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In LeaFTL, we set the 𝐿 as 1 byte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Thus, each segment can index 256 LPA-PPA mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We use a 16-bit floating point to store the value of the slope 𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' And the intercept 𝐼 of a segment can be represented in 4 bytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Therefore, in combination with 𝑆𝐿𝑃𝐴, both accurate and approximate segments can be encoded with 8 bytes (see Figure 6), which are memory aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL: A Learning-based Flash-Translation Layer for Solid-State Drives (a) Unoptimized learned segments (b) Optimized learned segments with sorting Learned Segments 78 32 33 76 Flush Data Buffer 115 34 38 Flash Block 78 32 33 76 115 34 38 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LPA 78 32 33 76 115 34 38 Learned Segments Flush Data Buffer Flash Block 32 33 34 38 76 78 115 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LPA 78 32 33 76 115 34 38 115 32 33 34 38 76 78 Figure 7: An example of reducing the number of learned seg- ments via exploiting the flash block allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL uses the least significant bit of the 𝐾 to indicate segment types (0 for accurate segments, 1 for approximate segments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' This has negligible impact on the address translation accuracy, because 𝐾 ∈ [0, 1], which will only affect the tenth digit after decimal point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='3 Improve the Learning Efficiency To further reduce the number of learned segments, LeaFTL performs optimizations to improve its learning efficiency of address mappings by exploiting the flash block allocation in SSD controllers, as shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Flash pages are usually buffered in the SSD controller and written to flash chips at a flash block granularity, for utilizing the internal bandwidth and avoiding the open-block problem [6, 22, 37, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' This allows LeaFTL to learn more space-efficient index segments (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=', index segments can cover more LPA-PPA mappings) by reordering the flash pages with their LPAs in the data buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As shown in Figure 7 (a), LeaFTL learns 5 index segments (78), (32, 33), (76), (115), and (34, 38) with 𝛾 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' After sorting the pages in the data buffer shown in Figure 7 (b), LeaFTL generates 3 index segments (32, 33, 34, 38), (76, 78), and (115).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' To develop the optimized learned segments, LeaFTL sorts the flash pages in ascending order of their LPAs in the data buffer (8MB by default).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' When pages in the data buffer is flushed to the flash chips, their PPAs are in ascending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' This ensures a mono- tonic address mapping between LPAs and PPAs, which reduces the number of index segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='4 Manage Learned Index Segments Upon new data updates or GC in the SSD, the learned index seg- ments need to be updated, due to the intrinsic property (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=', out-of- place update) of SSDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Unfortunately, the direct updates to learned index segments are expensive, since we have to relearn the in- dex segments with new PPAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' This relearning procedure not only consumes extra compute cycles, but also involves additional flash accesses, since we have to access the corresponding flash pages to obtain accurate PPAs for some of the LPAs in the index segment being updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' For instance, for in-place update to an approximate Level 0 Level 1 0 63 100 200 230 255 16 127 206 240 non-overlapping at each level segments can overlap across levels Figure 8: The learned index segments are managed in a log- structured manner in LeaFTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' segment, it can incur 21 flash accesses on average when relearn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In-place update also breaks the existing LPA-to-PPA mapping patterns, which results in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='2× additional segments and memory footprint, according to our experiments with various workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' To address this challenge, we manage the learned index segments in a log-structured manner, as shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Therefore, the newly learned index segments will be appended to the log structure (level 0 in Figure 8) and used to index the updated LPA-PPA map- pings, while the existing learned segments (level 1 and lower levels in Figure 8) can still serve address translations for LPAs whose map- pings have not been updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Such a structure supports concurrent lookups as enabled in the traditional log-structured merge tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As we insert the newly learned index segments at the top level of the log-structured tree, this minimizes the impact on other segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Log-Structured Mapping Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The log-structured mapping ta- ble has multiple levels to maintain the temporal order of index seg- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As discussed, the topmost level has the most recent learned index segments, and the lower level stores the older segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' For the segments on the same level, LeaFTL ensures that they are sorted and do not have overlapped LPAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' This is for fast location of the corresponding learned index segments in each level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' For the seg- ments across the levels, they may have overlapped LPAs, due to the nature of the log-structured organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' And the segments with overlapped LPA-PPA mappings will be compacted periodically for space reclamation (see its detailed procedure in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Manage Two Types of Index Segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL manages the ac- curate and approximate index segments in the same log-structured mapping table, as they can be encoded in the same format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' For each accurate segment, we can directly infer its indexed LPAs with the 𝑆𝐿𝑃𝐴, 𝐾, and 𝐿, since it has a regular pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' However, for approx- imate index segments, we only have the knowledge of the starting LPA and the end LPA with 𝑆𝐿𝑃𝐴 + 𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Its encoded LPAs cannot be directly inferred from their metadata (𝑆𝐿𝑃𝐴, 𝐿, 𝐾, 𝐼), since they are learned from irregular access patterns and may have mispredictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' If two approximate segments have overlapping LPA ranges, we could obtain inaccurate PPAs from the learned index segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As shown in Figure 9 (a), given an LPA with the value 105, we will check the segment at Level 0 and may get an inaccurate PPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' This will also affect the efficiency of the segment compaction, with which we eliminate duplicated entries between segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' To address this challenge, LeaFTL uses a Conflict Resolution Buffer (CRB) for each LPA group to store the LPAs indexed by each approximate segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The main purpose of CRB is to help LeaFTL check whether a given LPA belongs to one approximate segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The CRB is a nearly-sorted list [10] by the starting LPAs of its ap- proximate segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' To be specific, the CRB ensures the following Jinghan Sun, Shaobo Li, Yunxin Sun, Chao Sun, Dejan Vucinic, and Jian Huang 100 6 K1 I1 [100, 101, 103, 104, 106] 102 6 K2 I2 [102, 105, 107, 108] L0 L1 LPAs Lookup (LPA = 105) (a) Approximate index segments that index overlapped LPAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Conflict Resolution Buffer 100 101 103 104 106 null 102 105 107 108 null .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Lookup (LPA = 105) 102 6 K2 I2 (b) Resolve the conflict between approximate segments with CRB Figure 9: A case study of conflict resolution buffer for ap- proximate learned index segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' properties: (1) the LPAs belong to the same approximate segment are stored contiguously;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' (2) different approximate segments are sorted by their starting LPA, and CRB uses a 𝑛𝑢𝑙𝑙 byte to separate these segments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' (3) it does not have redundant LPAs, which means an LPA will appear at most once in the CRB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' This is achieved by removing existing same LPAs when we insert new approximate segments into the CRB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' However, if the 𝑆𝐿𝑃𝐴 of a new approximate segment is the same as any starting LPAs that have been stored in the CRB, LeaFTL will update the 𝑆𝐿𝑃𝐴 of the old segment with the adjacent LPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Take Figure 9 (b) as an example, upon a new approximate segment with 𝑆𝐿𝑃𝐴 = 100, we will update the 𝑆𝐿𝑃𝐴 of the existing segment to 101, and then insert the new segment into the CRB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In this case, LeaFTL will ensure each approximate segment will have its unique 𝑆𝐿𝑃𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' This will facilitate the approximate LPA-PPA address translation with high accuracy confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Since CRB is nearly sorted, its insertion, deletion, and lookup operations are fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The CRB is also space efficient, as each LPA (the offset in its corresponding LPA group) will take only one byte, and it guarantees that there are no redundant LPAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Therefore, the CRB will maximally store 256 LPAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Our experiments with a variety of storage workloads show that the CRB will take 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='9 bytes on average, as shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Given an LPA, in order to identify which approximate index segment it belongs to, LeaFTL will check the CRB with binary search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Once the LPA is found, LeaFTL will search to its left until identifying the 𝑆𝐿𝑃𝐴, and this 𝑆𝐿𝑃𝐴 will be the starting LPA of the corresponding approximate segment, as shown in Figure 9 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Therefore, CRB can assist LeaFTL to resolve the LPA lookups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='5 Handle Address Misprediction As discussed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='2, the mapping table entries encoded with ap- proximate segments may occasionally incur mispredictions and return an approximated PPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' These approximate segments have a guaranteed error bound [−𝛾,𝛾], where 𝛾 is a constant value that can be specified in the linear regression algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' To verify the correctness of the address translation, a simple method is to access MSR-hm MSR-src2 MSR-prxy MSR-prn MSR-usr FIU-home FIU-mail 0 100 200 300 CRB Size (in Bytes) Average 99 Percentile Figure 10: The distribution of CRB sizes for different storage workloads, when we set 𝛾 = 4 in LeaFTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' PPA1 PPA2 PPA3 PPA4 PPA5 Data Blocks Data OOB Flash Page LPA2 LPA4 LPA Reverse Mapping Figure 11: The out-of-band (OOB) metadata organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' It stores the reverse mapping for its neighbor PPAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' the flash page with the predicted PPA, and use the reverse mapping (its corresponding LPA) stored in the OOB metadata of the flash page to check whether the LPA matches or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In this case, upon a PPA misprediction, we need log(𝛾) flash accesses on average to identify the correct PPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' To avoid extra flash accesses for address mispredictions, LeaFTL leverages the OOB of the flash page to store the reverse mappings of its neighbor PPAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' This is developed based on the insight that: with a 𝑃𝑃𝐴𝑙𝑒𝑎𝑟𝑛𝑒𝑑 obtained from an approximate segment, its er- ror bound [−𝛾,𝛾] guarantees that the correct PPA is in the range of [𝑃𝑃𝐴𝑙𝑒𝑎𝑟𝑛𝑒𝑑 − 𝛾, 𝑃𝑃𝐴𝑙𝑒𝑎𝑟𝑛𝑒𝑑 + 𝛾], as discussed in Figure 4 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Thus, upon a misprediction, LeaFTL will read the flash page with 𝑃𝑃𝐴𝑙𝑒𝑎𝑟𝑛𝑒𝑑, and use its OOB to find the correct PPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In this case, LeaFTL ensures that it will incur only one extra flash access for address mispredictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' This is a feasible approach, as the OOB size is usually 128–256 bytes in modern SSDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As each LPA takes 4 bytes, we can store 32–64 reverse mapping entries in the OOB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We show the OOB organization of LeaFTL in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' For the flash page 𝑃𝑃𝐴𝑋 , the first 2𝛾 + 1 entries in its OOB correspond to the LPAs for the flash pages [𝑃𝑃𝐴𝑋 − 𝛾, 𝑃𝑃𝐴𝑋 + 𝛾].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' For the flash pages at the beginning and end of a flash block, we may not be able to obtain the reverse mapping of their neighbor PPAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We place the 𝑛𝑢𝑙𝑙 bytes in the corresponding entry of the OOB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='6 Preserve Other Core FTL Functions LeaFTL preserves the core functions such as GC and wear leveling in an FTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' It follows the same GC and wear leveling policies in modern SSDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' When the number of free blocks in an SSD is below a threshold (usually 15-40% of the total flash blocks), the SSD con- troller will trigger the GC execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL employs the greedy algorithm [5] to select the candidate blocks which have the minimal LeaFTL: A Learning-based Flash-Translation Layer for Solid-State Drives ALGORITHM 1: LeaFTL operations Input: 𝑔𝑟𝑜𝑢𝑝𝑠 ← 𝐿𝑒𝑎𝐹𝑇𝐿 𝑔𝑟𝑜𝑢𝑝 𝑝𝑎𝑟𝑡𝑖𝑡𝑖𝑜𝑛𝑠 // Insert/Update Segment in the LeaFTL 1 Function 𝑠𝑒𝑔_𝑢𝑝𝑑𝑎𝑡𝑒(𝑠𝑒𝑔𝑚𝑒𝑛𝑡,𝑙𝑒𝑣𝑒𝑙): 2 𝑠𝑒𝑔_𝑝𝑜𝑠 = 𝑏𝑖𝑛𝑎𝑟𝑦_𝑠𝑒𝑎𝑟𝑐ℎ(𝑙𝑒𝑣𝑒𝑙,𝑠𝑒𝑔𝑚𝑒𝑛𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='𝑆𝐿𝑃𝐴) 3 𝑙𝑒𝑣𝑒𝑙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='𝑖𝑛𝑠𝑒𝑟𝑡 (𝑠𝑒𝑔𝑚𝑒𝑛𝑡,𝑠𝑒𝑔_𝑝𝑜𝑠) 4 if 𝑛𝑜𝑡 𝑠𝑒𝑔𝑚𝑒𝑛𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='𝑎𝑐𝑐𝑢𝑟𝑎𝑡𝑒 then 5 Insert LPAs into CRB and remove redundant LPAs 6 if 𝑠𝑒𝑔𝑚𝑒𝑛𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='𝑆𝐿𝑃𝐴 exists in CRB then 7 Update the 𝑆𝐿𝑃𝐴 of the old segment 8 𝑣𝑖𝑐𝑡𝑖𝑚_𝑠𝑒𝑔𝑚𝑒𝑛𝑡𝑠 ← All segments that overlap the 𝑠𝑒𝑔𝑚𝑒𝑛𝑡 starting with 𝑠𝑒𝑔_𝑝𝑜𝑠 9 foreach 𝑣𝑖𝑐𝑡𝑖𝑚 ∈ 𝑣𝑖𝑐𝑡𝑖𝑚_𝑠𝑒𝑔𝑚𝑒𝑛𝑡𝑠 do 10 𝑠𝑒𝑔_𝑚𝑒𝑟𝑔𝑒 (𝑠𝑒𝑔𝑚𝑒𝑛𝑡, 𝑣𝑖𝑐𝑡𝑖𝑚) // if marked as removable by seg_merge() 11 if 𝑣𝑖𝑐𝑡𝑖𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='𝐿 = −1 then 12 𝑙𝑒𝑣𝑒𝑙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='𝑟𝑒𝑚𝑜𝑣𝑒 (𝑣𝑖𝑐𝑡𝑖𝑚) 13 if 𝑠𝑒𝑔𝑚𝑒𝑛𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='𝑜𝑣𝑒𝑟𝑙𝑎𝑝𝑠 (𝑣𝑖𝑐𝑡𝑖𝑚) then 14 Pop 𝑣𝑖𝑐𝑡𝑖𝑚 to the next level 15 if 𝑣𝑖𝑐𝑡𝑖𝑚 has overlaps in the next level then 16 Create level for 𝑣𝑖𝑐𝑡𝑖𝑚 to avoid recursion // Lookup LPA in the LeaFTL 17 Function 𝑙𝑜𝑜𝑘𝑢𝑝(𝑙𝑝𝑎): 18 foreach 𝑙𝑒𝑣𝑒𝑙 ∈ 𝑔𝑟𝑜𝑢𝑝𝑠 [𝑙𝑝𝑎 𝑚𝑜𝑑 256] do 19 𝑠𝑒𝑔_𝑝𝑜𝑠 = 𝑏𝑖𝑛𝑎𝑟𝑦_𝑠𝑒𝑎𝑟𝑐ℎ(𝑙𝑒𝑣𝑒𝑙,𝑙𝑝𝑎) 20 𝑠𝑒𝑔𝑚𝑒𝑛𝑡 = 𝑙𝑒𝑣𝑒𝑙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='𝑔𝑒𝑡_𝑠𝑒𝑔𝑚𝑒𝑛𝑡 (𝑠𝑒𝑔_𝑝𝑜𝑠) 21 if ℎ𝑎𝑠_𝑙𝑝𝑎(𝑠𝑒𝑔𝑚𝑒𝑛𝑡, 𝑙𝑝𝑎) then 22 return 𝑠𝑒𝑔𝑚𝑒𝑛𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑒𝑃𝑃𝐴(𝑙𝑝𝑎) // LeaFTL Compaction 23 Function 𝑠𝑒𝑔_𝑐𝑜𝑚𝑝𝑎𝑐𝑡(): 24 foreach 𝑔𝑟𝑜𝑢𝑝 ∈ 𝑔𝑟𝑜𝑢𝑝𝑠 do 25 foreach 𝑢𝑝𝑝𝑒𝑟_𝑙𝑒𝑣𝑒𝑙,𝑙𝑜𝑤𝑒𝑟_𝑙𝑒𝑣𝑒𝑙 ∈ 𝑔𝑟𝑜𝑢𝑝 do 26 foreach 𝑠𝑒𝑔𝑚𝑒𝑛𝑡 ∈ 𝑢𝑝𝑝𝑒𝑟_𝑙𝑒𝑣𝑒𝑙 do 27 𝑠𝑒𝑔_𝑢𝑝𝑑𝑎𝑡𝑒 (𝑠𝑒𝑔𝑚𝑒𝑛𝑡,𝑙𝑜𝑤𝑒𝑟_𝑙𝑒𝑣𝑒𝑙) 28 if 𝑢𝑝𝑝𝑒𝑟_𝑙𝑒𝑣𝑒𝑙 is empty then 29 𝑔𝑟𝑜𝑢𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='𝑟𝑒𝑚𝑜𝑣𝑒 (𝑢𝑝𝑝𝑒𝑟_𝑙𝑒𝑣𝑒𝑙) number of valid pages, for reducing the data movement overhead at GC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As the GC move the valid pages from the candidate blocks to the free blocks, LeaFTL places these valid pages into the DRAM buffer, sort them by their LPAs, and learn a new index segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The learning procedure is the same as we build index segments for new flash writes/updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Thus, the address mapping of the valid pages is updated after the GC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL also ensures all the flash blocks age at the same rate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=', wear leveling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' It uses the throttling and swapping mechanism developed in existing GC, in which the cold data blocks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=', blocks not frequently accessed) will be migrated to hot blocks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=', blocks that experience more wear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL will learn new indexes for these swapped blocks and insert them into the mapping table to update their address mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='7 LeaFTL Operations Now we describe the LeaFTL operations, including segment cre- ation, insert/update, LPA lookup, and compaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We discuss their procedures, and use examples to illustrate each of them, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We present their detailed procedures in Algorithm 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' ALGORITHM 2: Segment Merge // Check if Segment Contains LPA 1 Function ℎ𝑎𝑠_𝑙𝑝𝑎(𝑠𝑒𝑔, 𝑙𝑝𝑎): 2 𝑎𝑐𝑐 ← 𝑠𝑒𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='𝑎𝑐𝑐𝑢𝑟𝑎𝑡𝑒 3 if 𝑙𝑝𝑎 ∉ [𝑠𝑒𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='𝑆𝐿𝑃𝐴,𝑠𝑒𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='𝑆𝐿𝑃𝐴 + 𝑠𝑒𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='𝐿] 𝑜𝑟 (𝑛𝑜𝑡 𝑎𝑐𝑐 & 𝑐ℎ𝑒𝑐𝑘 (𝐶𝑅𝐵) 𝑓 𝑎𝑖𝑙𝑒𝑑) 𝑜𝑟 (𝑎𝑐𝑐 & (𝑙𝑝𝑎 − 𝑠𝑒𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='𝑆𝐿𝑃𝐴) 𝑚𝑜𝑑 ⌈ 1 𝑠𝑒𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='𝐾 ⌉ ≠ 0) then 4 𝑟𝑒𝑡𝑢𝑟𝑛 𝐹𝑎𝑙𝑠𝑒 5 𝑟𝑒𝑡𝑢𝑟𝑛 𝑇𝑟𝑢𝑒 // Convert Segment into a Temporary Bitmap 6 Function 𝑔𝑒𝑡_𝑏𝑖𝑡𝑚𝑎𝑝(𝑠𝑒𝑔, 𝑠𝑡𝑎𝑟𝑡, 𝑒𝑛𝑑): 7 𝑏𝑚 ← 𝑏𝑖𝑡𝑚𝑎𝑝 𝑜𝑓 𝑙𝑒𝑛𝑔𝑡ℎ (𝑒𝑛𝑑 − 𝑠𝑡𝑎𝑟𝑡 + 1) 8 foreach 𝑙𝑝𝑎 ∈ [𝑠𝑡𝑎𝑟𝑡,𝑒𝑛𝑑] do 9 if ℎ𝑎𝑠_𝑙𝑝𝑎(𝑠𝑒𝑔, 𝑙𝑝𝑎) then 10 𝑏𝑚[𝑙𝑝𝑎 − 𝑠𝑡𝑎𝑟𝑡 ] = 1 11 else 12 𝑏𝑚[𝑙𝑝𝑎 − 𝑠𝑡𝑎𝑟𝑡 ] = 0 13 return 𝑏𝑚 // Merge a New Segment with an Old Segment 14 Function 𝑠𝑒𝑔_𝑚𝑒𝑟𝑔𝑒(𝑛𝑒𝑤, 𝑜𝑙𝑑): 15 𝑠𝑡𝑎𝑟𝑡 ← 𝑚𝑖𝑛(𝑛𝑒𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='𝑆𝐿𝑃𝐴, 𝑜𝑙𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='𝑆𝐿𝑃𝐴) 16 𝑒𝑛𝑑 ← 𝑚𝑎𝑥 (𝑛𝑒𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='𝑆𝐿𝑃𝐴 + 𝑛𝑒𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='𝐿, 𝑜𝑙𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='𝑆𝐿𝑃𝐴 + 𝑜𝑙𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='𝐿) 17 𝑏𝑚𝑛𝑒𝑤 ← 𝑔𝑒𝑡_𝑏𝑖𝑡𝑚𝑎𝑝 (𝑛𝑒𝑤, 𝑠𝑡𝑎𝑟𝑡, 𝑒𝑛𝑑) 18 𝑏𝑚𝑜𝑙𝑑 ← 𝑔𝑒𝑡_𝑏𝑖𝑡𝑚𝑎𝑝 (𝑜𝑙𝑑, 𝑠𝑡𝑎𝑟𝑡, 𝑒𝑛𝑑) 19 𝑏𝑚𝑜𝑙𝑑 ← 𝑏𝑚𝑜𝑙𝑑 & ¬𝑏𝑚𝑛𝑒𝑤 20 𝑓 𝑖𝑟𝑠𝑡, 𝑙𝑎𝑠𝑡 ← the first and last valid bit of 𝑏𝑚𝑜𝑙𝑑 21 𝑜𝑙𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='𝑆𝐿𝑃𝐴, 𝑜𝑙𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='𝐿 ← 𝑓 𝑖𝑟𝑠𝑡 + 𝑠𝑡𝑎𝑟𝑡, 𝑙𝑎𝑠𝑡 − 𝑓 𝑖𝑟𝑠𝑡 22 if no valid bits in 𝑜𝑙𝑑 then 23 𝑜𝑙𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='𝐿 ← −1 // mark it as removable 24 if 𝑛𝑜𝑡 𝑜𝑙𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='𝑎𝑐𝑐𝑢𝑟𝑎𝑡𝑒 then 25 Remove outdated LPAs in CRB Creation of Learned Segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Once the data buffer of the SSD controller is filled, LeaFTL takes the LPAs and PPAs of the flash pages in the buffer as the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' It sorts the LPA-PPA mappings by reordering the flash pages with their LPAs (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='3), and uses greedy piecewise linear regression [64] to learn the index segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Insert/Update of Learned Segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' When we insert or update a new learned index segment, we will place it in the topmost level of the log-structured mapping table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Since each level of the map- ping table is sorted, we can quickly identify its insert location via a binary search (line 2 in Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' If the new segment is ap- proximate, LeaFTL will update the CRB for future lookups (line 4-7 in Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' After that, LeaFTL will check whether the new segment overlaps with existing segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' If yes, LeaFTL will identify the overlapped LPAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The overlap detection is performed by the comparison between the LPA range of the new segment and [𝑆𝐿𝑃𝐴,𝑆𝐿𝑃𝐴 +𝐿] of the adjacent segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We group these overlap- ping segments as a list of victim segments (line 8 in Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL will merge segments to remove outdated LPAs (line 10 in Algorithm 1 and line 14-25 in Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' To fulfill the segment merge, LeaFTL will use the 𝑆𝐿𝑃𝐴, 𝐿, and 𝐾 to reconstruct the list of the encoded LPAs in the victim segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' And it will create a bitmap to index these encoded LPAs (line 6-13 in Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Given an accurate segment with 𝑆𝐿𝑃𝐴 = 100, 𝐾 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='5, 𝐿 = 6, we can infer that its encoded LPAs are [100, 102, 104, 106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We can transfer the LPA list to the bitmap [1010101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' If the victim Jinghan Sun, Shaobo Li, Yunxin Sun, Chao Sun, Dejan Vucinic, and Jian Huang MSR-hm MSR-src2 MSR-prxy MSR-prn MSR-usr FIU-home FIU-mail 0 5 10 15 20 # of Levels in Each Group Average 99 Percentile Figure 12: A study of the number of levels in the log- structured mapping table for different storage workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' L0 0 63 T0 Initial Snapshot T1 Update LPAs 200 - 255 L0 0 63 200 255 T2 Update LPAs 16 - 31 L0 16 31 200 255 L1 0 63 T4 Update [72,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 73,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 80] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='L0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='255 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='L1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='T6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='Lookup LPA 78 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='L0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='L1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='T8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='Compaction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='Timeline ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='Segments ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='CRB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='T7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='Update LPAs 32 - 90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='82 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='72 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='255 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='82 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='72 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='T5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='Lookup LPA 50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='L0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='L1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='255 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='82 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='72 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='L0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='L1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='255 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='82 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='L0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='255 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='Start ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='End ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='Accurate Segment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='Start ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='End ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='Approximate Segment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='72 73 80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='/ 75 78 82 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='72 73 80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='75 78 82 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='72 73 80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='75 78 82 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='75 78 82 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='T3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='Update [75,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 78,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 82] L0 16 31 200 255 L1 0 63 75 82 75 78 82 Figure 13: Examples that involve update/insert,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' lookup,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' and compaction operations in LeaFTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' segment is an approximate segment, LeaFTL will leverage the 𝑆𝐿𝑃𝐴, 𝐿, and the LPAs stored in the CRB to reconstruct the encoded LPAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Afterwards, LeaFTL will conduct a comparison between the bitmaps to identify the overlapped LPAs (line 15-19 in Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' During the segment merge, LeaFTL will update the 𝑆𝐿𝑃𝐴 and 𝐿 of the old segments accordingly, remove the outdated LPAs from CRB for approximate segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Note that we do not update the 𝐾 and 𝐼 for the victim segments during the merge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' After the merge, (1) if the victim segment does not contain any valid LPA (𝐿 is negative), it will be removed from the mapping table (line 11-12 in Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' (2) If the victim segment has valid LPAs but their range still overlaps with the new segment, the victim segment will be moved to the next level in the log- structured mapping table (line 13-16 in Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' To avoid recursive updates across the levels, we create a new level for the victim segment if it also overlaps with segments in the next level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' According to our study of diverse workloads, this will not create many levels in the mapping table (see Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' (3) If the victim segment has valid LPAs and they do not overlap with the new segment, we do not need to perform further operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' This is because the victim segment is updated with new 𝑆𝐿𝑃𝐴 and 𝐿 during segment merge (line 20-25 in Algorithm 2), and the new segment insertion keeps each level sorted (line 3 in Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' To facilitate our discussion, we present a few examples in Fig- ure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' At the initial stage, the mapping table has one segment that indexes the LPA range [0, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' At 𝑇1, the new segment [200, 255] is directly inserted into the topmost level, as it does not overlap with existing segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' At 𝑇2, we insert a new segment [16, 31] that has overlaps with the old segment [0, 63], LeaFTL conducts the segment merge procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' After that, the old segment still has valid LPAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Thus, it moves to level 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' At 𝑇3 and 𝑇4, we insert two approximate segments [75, 82] and [72, 80], LeaFTL will also insert their encoded LPAs into the CRB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The segment [75, 82] will be moved to the next level as it overlaps with the new segment [72, 80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LPA Lookup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL conducts an LPA lookup from the top- most level of the mapping table with binary searches (line 19 in Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We will check whether the LPA is represented by the matched segment (line 21 in Algorithm 1, line 1-5 in Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' If the 𝐿𝑃𝐴 ∈ [𝑆𝐿𝑃𝐴,𝑆𝐿𝑃𝐴 + 𝐿] of the segment, LeaFTL will check the least bit of its 𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' If the least bit of 𝐾 is 0, it is an accurate segment, and LeaFTL will use 𝑓 (𝐿𝑃𝐴) = ⌈𝐾 ∗ 𝐿𝑃𝐴 + 𝐼⌉ to get the accurate PPA (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Otherwise, it is an approximate segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL will check the CRB to identify the 𝑆𝐿𝑃𝐴 of the segment, following the approach described in Figure 9 and §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL will use the same 𝑓 (𝐿𝑃𝐴) formula to obtain the PPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' If the LPA is not found in the top level of the mapping table, LeaFTL will search the lower levels until a segment is identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We use Figure 13 to illustrate the lookup procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' At 𝑇5, we conduct the address translation for 𝐿𝑃𝐴 = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' However, none of the segments in the level 0 covers this LPA, LeaFTL will continue the search in the level 1 and find the accurate segment [0, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' At 𝑇6, we do the address translation for 𝐿𝑃𝐴 = 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL finds that the LPA 78 is in the LPA range of the segment [72, 80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Since this is an approximate segment, LeaFTL checks the CRB and finds this LPA is actually indexed by the segment [75, 82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' With the PPA, LeaFTL will read the corresponding flash page and use the reversed mapping (its corresponding LPA) in its OOB to ver- ify the correctness of the address translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Upon mispredictions, we will use the approach discussed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='5 to handle it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Segment Compaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The purpose of the compaction is to merge segments with overlapped LPAs across different levels, which further saves memory space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL will iteratively move the upper- level segments into the lower level, until the mapping table is fully compacted (line 27 in Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' When an approximate segment is removed, its corresponding CRB entries will also be deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As shown in 𝑇7 of Figure 13, we insert a new segment [32, 90] which fully covers the LPA range of the segment [72, 80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' After merge, LeaFTL removes the old segment [72, 80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' However, some segments LeaFTL: A Learning-based Flash-Translation Layer for Solid-State Drives Conflict Resolution Buffer (CRB) Key Data Structures in LeaFTL 6 Log-Structured Mapping Table 5 L0 L1 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Group 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' CRB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 0 63 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 16 31 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 64 95 Figure 14: Key data structures used in LeaFTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' in the level 0 still overlap with the segments in the level 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' After 𝑇8, LeaFTL will remove outdated segments and LPAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL performs segment compaction after each 1 million writes by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' According to our experiments with various storage work- loads, the segment compaction of the entire mapping table will take 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='1 milliseconds (the time of 20-40 flash writes) on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Consider the low frequency (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=', once per 1 million writes), the compaction incurs trivial performance overhead to storage operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='8 Put It All Together LeaFTL is compatible with existing FTL implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As shown in Figure 14, it uses the log-structured mapping table ( 5 ) to replace the address mapping cache ( 1 in Figure 3), and employs CRB ( 6 ) for assisting the address translation of approximate segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The CRB requires trivial storage space in the SSD DRAM (see Figure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Read Operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' For a read request, LeaFTL will first check the data cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' For a cache hit, LeaFTL serves the read request with the cached flash page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Otherwise, LeaFTL will perform address translation with 5 (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' If there is a misprediction of PPA, LeaFTL checks the OOB of the mispredicted flash page, read the correct page (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='5), and updates the data cache with the page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Write Operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' For a write request, LeaFTL buffers it in the data cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Once the buffered writes reach the size of a flash block, LeaFTL will allocate a free block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' It will sort the writes in the buffer based on their LPAs, and learn new index segments with the PPAs of the allocated flash block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' This enables LeaFTL to group more LPA- PPA mappings in the same index segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' After that, LeaFTL will insert the new index segment in the mapping table, and flush the buffered data to the flash blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' For those writes, LeaFTL will also check whether their LPAs exist in the mapping table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' If yes, LeaFTL will update their corresponding entries in 3 BVC and 4 PVT to indicate that they become invalid and can be garbage collected in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Otherwise, the new learned segments will have their LPA-PPA mappings for future address translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL caches the mapping table in SSD DRAM for fast lookup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The table will also be stored in the flash blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL utilizes the existing 2 GMD to index the translation pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' If a segment is not found in the cached mapping table, LeaFTL will fetch it from the translation blocks and place it in the cached mapping table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Crash Consistency and Recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Upon system crashes or power failures, LeaFTL guarantees the crash consistency of learned in- dexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In order to ensure the data durability of DRAM buffer in SSD controllers, modern SSDs today have employed battery-backed DRAM and power loss protection mechanisms [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' With battery- backed DRAM, LeaFTL has sufficient time to persist the up-to-date mapping table to the flash blocks and record their PPAs in the GMD Table 1: SSD configurations in our simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Parameter Value Parameter Value Capacity 2TB #Channels 16 Page size 4KB OOB size 128B DRAM size 1GB Pages/block 256 Read latency 20𝜇s Write latency 200𝜇s Erase 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='5 millisecs Overprovisioning ratio 20% ( 2 in Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' During the data recovery, LeaFTL reads the GMD to locate its mapping table and place it into the DRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Without battery-backed DRAM, LeaFTL periodically flushes the learned mapping table and the Block Validity Counter ( 3 BVC in Figure 3) into the flash blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' When GC is triggered, LeaFTL also flushes the updated mapping table and BVC into the flash blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Upon crashes, LeaFTL will scan all the flash blocks at the channel- level parallelism, and reconstruct an up-to-date BVC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL is able to identify the flash blocks allocated since the last mapping table flush, by comparing the up-to-date BVC with the stored BVC in the SSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Therefore, LeaFTL only needs to relearn the index segments for these recently allocated flash blocks and add them into the mapping table (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='9 Implementation Details SSD Simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We implement LeaFTL based on a trace-driven simulator WiscSim [27], which has provided an event simulation environment for the end-to-end performance analysis of SSDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We extend WiscSim by implementing an LRU-based read-write cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL also preserves the functions of existing FTL, such as GC and wear-leveling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' To support the learned indexing, LeaFTL employs a simple linear regression algorithm [65], which incurs negligible computation overhead with modern storage processors (see §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The error bound 𝛾 for learned segments is configurable, and we set it to 0 by default in LeaFTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' SSD Prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We also develop a real system prototype with an open-channel SSD to validate the functions and efficiency of LeaFTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The SSD has 1TB storage capacity with 16 KB flash page size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' It has 16 channels, each channel has 16K flash blocks, and each flash block has 256 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' It enables developers to implement their own FTL in the host by providing basic I/O commands such as read, write, and erase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We implement LeaFTL with 4,016 lines of code using C programming language with the SDK library of the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 4 EVALUATION Our evaluation shows that: (1) LeaFTL significantly reduces the address mapping table size, and the saved memory brings perfor- mance benefits (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' (2) the benefits of LeaFTL are validated on a real SSD device (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' (3) LeaFTL can achieve additional memory savings and performance benefits with larger error-tolerance, and it demonstrate generality for different SSD configurations (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' (4) Its learning procedure does not introduce much extra overhead to the SSD controller (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' (5) It has minimal negative impact on the SSD lifetime (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Jinghan Sun, Shaobo Li, Yunxin Sun, Chao Sun, Dejan Vucinic, and Jian Huang Table 2: Real workloads used in our real SSD evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Workload Description OLTP [59] Transactional benchmark in the FileBench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' CompFlow (CompF) [59] File accesses in a computation flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' TPCC [13] Online transaction queries in warehouses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' AuctionMark (AMark) [13] Activity queries in an auction site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' SEATS [13] Airline ticketing system queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' MSR-hm MSR-src2 MSR-prxy MSR-prn MSR-usr FIU-home FIU-mail 50x 20x 10x 5x 2x 1x Memory Footprint Reduction DFTL SFTL LeaFTL Figure 15: The reduction on the mapping table size of LeaFTL, in comparison with DFTL and SFTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='1 Experiment Setup We examine the efficiency of LeaFTL with both the SSD simula- tor and real SSD prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As for the evaluation with the SSD simulator, we configure a 2TB SSD with 4KB flash pages and 1GB DRAM in the SSD controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We list the core SSD parameters in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' For other parameters, we use the default setting in the WiscSim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We use a variety of storage workloads that include the block I/O traces from enterprise servers from Microsoft Research Cambridge [45] and workload traces from computers at FIU [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As for the evaluation with the real SSD prototype (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='9), we validate the benefits of LeaFTL using a set of real-world file system benchmarks and data intensive applications as shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Before we measure the performance, we run a set of workloads consisting of various real-world and synthetic storage workload traces to warm up the SSD and make sure the GC will be executed during the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We compare LeaFTL with state-of-the-art page-level mapping schemes described as follows 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' DFTL (Demand-based FTL) [20]: it uses a page-level mapping scheme, and caches the most recently used address translation entries in the SSD DRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' SFTL (Spatial-locality-aware FTL) [25]: it is a page-level map- ping that exploits the spatial locality and strictly sequential access patterns of workloads to condense mapping table entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='2 Memory Saving and Performance We first evaluate the benefits of LeaFTL on the memory saving and storage performance with the SSD simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As shown in Figure 15, LeaFTL reduces the mapping table size by 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='5–37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='7×, compared to the page-level mapping scheme DFTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' This is because LeaFTL can group a set of page-level mapping entries into an 8- byte segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In comparison with SFTL, LeaFTL achieves up to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='3× (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='9× on average) reduction on the address mapping table for different storage workloads, when we set its 𝛾 = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=', the learned 1We do not compare LeaFTL with block-level and hybrid-level mappings, as they perform dramatically worse than the page-level mapping [20, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' MSR-hm MSR-src2 MSR-prxy MSR-prn MSR-usr FIU-home FIU-mail 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='0 Normalized Perf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' DFTL SFTL LeaFTL (a) SSD performance when using its DRAM mainly for the address mapping table (lower is better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' MSR-hm MSR-src2 MSR-prxy MSR-prn MSR-usr FIU-home FIU-mail 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='0 Normalized Perf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' DFTL SFTL LeaFTL (b) SSD performance when using its DRAM partially (up to 80%) for the address mapping table (lower is better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Figure 16: Performance improvement with LeaFTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' SEATS AMark TPCC OLTP CompF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='0 Normalized Perf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' DFTL SFTL LeaFTL Figure 17: Performance on the real SSD prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='9% 99% 90% 60% 30% 0% Percentage of Storage Accesses 100 101 102 103 Latency ( s) DFTL SFTL LeaFTL Figure 18: The latency distribution of storage accesses when running OLTP workload on the real SSD prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' segments are 100% accurate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' This is because LeaFTL captures more LPA-PPA mapping patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We now evaluate the performance benefit of LeaFTL from its saved memory space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We evaluate LeaFTL with two experimental settings: (1) the SSD DRAM is mainly used (as much as possible) for the mapping table;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' (2) the SSD DRAM is partially used for the mapping table, in which we ensure at least 20% of the DRAM will be used for the data caching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In the first setting, DRAM is almost used for mapping table in DFTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As shown in Figure 16 (a), LeaFTL reduces the storage access latency by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='6× on average (up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='7×), compared to SFTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' This is because LeaFTL saves more memory from the mapping table LeaFTL: A Learning-based Flash-Translation Layer for Solid-State Drives MSR-hm MSR-src2 MSR-prxy MSR-prn MSR-usr FIU-home FIU-mail SEATS AMark TPCC OLTP CompF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='0 Memory Footprint Reduction =0 =1 =4 =16 SSD Simulator Real SSD Figure 19: The reduction of the mapping table size of LeaFTL with different 𝛾 (lower is better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' =0 =1 =4 =16 0% 20% 40% 60% 80% 100% Percentage of Segments Accurate Approximate Figure 20: The distribution of learned segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' than SFTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' SFTL slightly outperforms DFTL, because it reduces the mapping table size by compressing mapping entries with grouping strictly sequential data accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In the second setting, as shown in Figure 16 (b), LeaFTL obtains 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='4× (up to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='4×) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='6× (up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='9×) performance speedup, compared to SFTL and DFTL, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='3 Benefits on the Real SSD Prototype We validate the benefits of LeaFTL on the real SSD prototype with real workloads (see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' They include filesystem benchmark suite FileBench [59], and transactional database workloads from BenchBase [13, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' All these workloads run on the ext4 file system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' With FileBench, we run OLTP and CompFlow (CompF) workloads to read/write 10GB files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' With BenchBase, we run TPCC, Auction- Mark (AMark), and SEATS workloads on MySQL, and their data- base sizes are 10–30GB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' These database workloads will generate 37–230GB read traffic and 26–59GB write traffic to the SSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We allo- cate 256MB DRAM to host the mapping table (for different DRAM sizes, see our sensitivity analysis in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We present the performance benefit of LeaFTL in Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Across all workloads, LeaFTL obtains 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='4× performance speedup on average (up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='5×), compared to SFTL and DFTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Similar to our evaluation with the SSD simulator implementation, the per- formance benefit of LeaFTL comes from the memory saving from the address mapping table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' And LeaFTL demonstrates comparable performance improvement on real SSD devices, in comparison with the SSD simulator in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We also show the latency distribution of storage accesses in Figure 18, when running the OLTP workload on the real SSD prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In comparison with existing FTL schemes, LeaFTL does not increase the tail latency of storage accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' And the higher cache hit ratio of LeaFTL brings latency reduction for many storage accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='4 Sensitivity Analysis Vary the value of 𝛾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As we increase the value of 𝛾 from 0 to 16, the size of the learned mapping table is reduced, as shown in MSR-hm MSR-src2 MSR-prxy MSR-prn MSR-usr FIU-home FIU-mail SEATS AMark TPCC OLTP CompF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='0 Normalized Perf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' =0 =1 =4 =16 SSD Simulator Real SSD Figure 21: Performance with various 𝛾 (lower is better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 256MB 512MB 1024MB (a) Various DRAM size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='0 Normalized Perf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 4KB 8KB 16KB (b) Various flash page size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='0 Normalized Perf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' DFTL SFTL LeaFTL Figure 22: SSD performance with different DRAM capacity and flash page size (lower is better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL achieves 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='3× reduction on average (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='2× on the real SSD) with 𝛾 = 16, compared to that of 𝛾 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The saved memory with a larger 𝛾 is achieved by learning a wider range of LPAs into approximate segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' To further understand this, we profile the distribution of segments learned by LeaFTL with different values of 𝛾, as shown in Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' When 𝛾 = 0, all the segments are accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' When 𝛾 = 16, 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='5% of the learned segments are approximate on average, and LeaFTL delivers 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='3× improvement on storage performance (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='2× with workloads on the real SSD), in comparison with the case of 𝛾 = 0 (see Figure 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Vary the SSD DRAM capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We now conduct the sensitivity analysis of SSD DRAM by varying its capacity from 256MB to 1GB on the real SSD prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As shown in Figure 22 (a), LeaFTL always outperforms DFTL and SFTL as we vary the SSD DRAM capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As we increase the DRAM capacity, the storage workloads are still bottlenecked by the available memory space for the data caching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL can learn various data access patterns and significantly reduce the address mapping table size, the saved memory further benefits data caching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Vary the flash page size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In this experiment, we fix the number of flash pages, and vary the flash page size from 4KB to 16KB in the SSD simulator, as SSD vendors usually use larger flash pages for increased SSD capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We use the simulator for this study, since the flash page size of the real SSD is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As shown in Figure 22 (b), LeaFTL always performs the best in comparison with DFTL and SFTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As we increase the flash page size to 16KB, we can cache less number of flash pages with limited DRAM capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Thus, LeaFTL experiences a slight performance drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As we fix the total SSD Jinghan Sun, Shaobo Li, Yunxin Sun, Chao Sun, Dejan Vucinic, and Jian Huang 1 5 10 15 20 25 30 35 (a) Number of Levels 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='99% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='9% 99% 90% 0% Percentage of Lookups MSR-prn MSR-usr MSR-src2 MSR-hm MSR-prxy FIU-home FIU-mail 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='5 (b) LPA Lookup Overhead (%) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='99% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='9% 99% 90% 0% Percentage of Lookups SEATS CompF OLTP TPCC AMark Figure 23: Performance overhead of the LPA lookup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' MSR-hm MSR-src2 MSR-prxy MSR-prn MSR-usr FIU-home FIU-mail SEATS AMark TPCC OLTP CompF 0 5 10 15 20 Misprediction (%) =0 =1 =4 =16 SSD Simulator Real SSD Figure 24: Misprediction ratio of flash pages access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' capacity and vary the page size, LeaFTL outperforms SFTL by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='2× and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='1× for the page size of 8KB and 16KB, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='5 Overhead Source in LeaFTL We evaluate the overhead sources in LeaFTL in three aspects: (1) the performance overhead of the learning procedure in LeaFTL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' (2) the LPA lookup overhead in the learned segments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' and (3) the overhead caused by the address misprediction in LeaFTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We evaluate the performance of segment learning and address lookup on an ARM Cortex-A72 core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' This core is similar to the storage processor used in modern SSDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The learning time for a batch of 256 mapping entries is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='8–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='8 𝜇s (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As we learn one batch of index segments for every 256 flash writes, the learning overhead is only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='02% of their flash write latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In LeaFTL, the LPA lookup is 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='2–67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='5 ns, as the binary search of segments is fast and some segments can be cached in the processor cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The lookup time is slightly higher as we increase𝛾, due to the additional CRB accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We also profile the cumulative distribution function (CDF) of the number of levels to lookup for each LPA lookup, and present the results in Figure 23 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' For most of the tested workloads, 90% of the mapping table lookup can be fulfilled at the topmost level, and 99% of the lookups are within 10 levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Although MSR-prn workload requires more lookups than other workloads, it only checks 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='4 levels on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We also evaluate the performance overhead of the LPA lookup on the real SSD, and show the results in Figure 23 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The extra lookup overhead for each flash read is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='21% on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' And for 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='99% of all the lookups, the additional overhead is less than 1% of the flash access latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Table 3: Overhead source of LeaFTL with an ARM core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 𝛾 0 1 4 Learning (256 LPAs) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='8 𝜇s 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='8 𝜇s 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='8 𝜇s Lookup (per LPA) 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='2 ns 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='5 ns 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='5 ns LeaFTL also has low misprediction ratios with approximate seg- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' This is because LeaFTL can still learn accurate segments even if 𝛾 > 0, and not all entries in the approximate segments will result in misprediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As shown in Figure 24, most of the workloads achieve less than 10% misprediction ratio when 𝛾 = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We obtain similar misprediction ratio on the real SSD prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Note that each misprediction only incurs one flash read access with the help of our proposed OOB verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='6 Impact on SSD Lifetime The flash blocks of an SSD can only undergo a certain amount of writes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In this experiment, we use the write amplification factor (WAF, the ratio between the actual and requested flash writes) to evaluate the SSD lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The SSD will age faster if the WAF is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As shown Figure 25, the WAF of LeaFTL is comparable to DFTL and SFTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' DFTL has larger WAF in most workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' SFTL and LeaFTL occasionally flush translation pages to the flash blocks, but the cost is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 5 DISCUSSION Why Linear Regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Unlike deep neural networks, the lin- ear regression used in LeaFTL is simple and lightweight, which takes only a few microseconds to learn an index segment with embedded ARM processors available in modern SSD controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In addition, the linear regression algorithm has been well studied, and offers guaranteed error bounds for its learned results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL is the first work that uses learning techniques to solve a critical system problem (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=', address mapping) in SSDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Adaptivity of LeaFTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL focuses on the page-level address translation, its design and implementation will not be affected by the low-level flash memory organization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=', TLC/QLC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As we use TLC/QLC technique to further increase the SSD capacity, the address mapping issue will become more critical, since the SSD DRAM capacity does not scale well and becomes the bottleneck for caching address mappings and user data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Recovery of Learned Index Segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As discussed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='8, us- ing a battery or large capacitor to preserve and persist the cached segments upon failures or crashes will simplify the recovery pro- cedure significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In our real SSD prototype, we do not assume the battery-backed DRAM is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Thus, we follow the conven- tional recovery approach in modern SSDs [20, 23], and scan flash blocks in parallel by utilizing the channel-level parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' When we run real workloads like TPCC on the SSD prototype, we intentionally reboot the system after running the workload for a period of time (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='5-3 hours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We find that the system can recover in 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='8 minutes on average whenever the reboot happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' This is similar to the time of recovering the conventional page-level mapping table in DFTL [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' This is mostly caused by scanning the blocks in a channel (70MB/s per channel in our SSD prototype), and the time for reconstructing recently learned segments is rela- tively low (101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='3 milliseconds on average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We believe the recovery LeaFTL: A Learning-based Flash-Translation Layer for Solid-State Drives MSR-hm MSR-src2 MSR-prxy MSR-prn MSR-usr FIU-home FIU-mail SEATS AMark TPCC OLTP CompF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='5 Write Amplification DFTL SFTL LeaFTL SSD Simulator Real SSD Figure 25: Write amplification factor of LeaFTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' time is not much of a concern as the recovery does not happen frequently in reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' And the recovery can be accelerated as we increase the channel-level bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In addition, if an SSD can tolerate more data losses, we can still ensure the crash consistency by only loading the stored index segments from flash chips, which requires minimum recovery time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 6 RELATED WORK Address Translation for SSDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' A variety of FTL optimizations have been proposed [8, 12, 20, 25, 28, 34, 49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' These works ex- ploited the data locality of flash accesses to improve the cache efficiency of the mapping table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' However, most of them were devel- oped with human-driven heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' An alternative approach is to integrate application semantics into the FTL, such as content-aware FTL [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' However, they were application specific and required signif- icant changes to the FTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL is a generic solution and does not require application semantics in its learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Researchers proposed to integrate the FTL mapping table into the host [18, 23, 26, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Typi- cal examples include DFS [26], Nameless writes [66], FlashMap [23], and FlatFlash [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL is orthogonal to them and can be applied to further reduce their memory footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Machine Learning for Storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Recent studies have been using learning techniques to build indexes such as B-trees, log-structured merge tree, hashmaps, and bloom filters [11, 14, 15, 32, 33, 42] for in-memory datasets, identify optimal cache replacement and prefetching policies [40, 53, 56, 57], facilitate efficient storage har- vesting [52], and drive the development of software-defined stor- age [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL applies learning techniques to optimize the address mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' However, unlike existing optimizations [43, 63] such as learned page table for virtual memory that used deep neural net- works to learn the patterns, LeaFTL provides a lightweight solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' SSD Hardware Development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' For the recent SSD innovations [3, 17, 19, 47] like Z-SSD [55], KVSSD [35], and ZNS SSD [21], DRAM capacity and storage processor are still the main constraints in SSD controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' As we scale the storage capacity, the challenge with the address translation becomes only worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Researchers recently deployed hardware accelerators inside SSD controllers for near- data computing [36, 41, 54, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' We wish to extend LeaFTL with in-storage accelerators to deploy more powerful learning models as the future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 7 CONCLUSION We present a learning-based flash translation layer, named LeaFTL for SSDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LeaFTL can automatically learn different flash access patterns and build space-efficient indexes, which reduces the ad- dress mapping size and improves the caching efficiency in the SSD controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Our evaluation shows that LeaFTL improves the SSD performance by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='4× on average for a variety of storage workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank the anonymous reviewers for their helpful comments and feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' This work is partially supported by the NSF CAREER Award 2144796, CCF-1919044, and CNS-1850317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' REFERENCES [1] 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' A Closer Look At SSD Power Loss Protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='kingston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='com/ en/blog/servers-and-data-centers/ssd-power-loss-protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [2] 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Harnessing Microcontrollers to Deliver Intelligent SSD Power Management and PLP Capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='atpinc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='com/de/about/stories/microcontroller- SSD-power-loss-protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [3] 3D NAND – An Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='simms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='uk/tech-talk/3d-nand-overview/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [4] Ahmed Abulila, Vikram Sharma Mailthoday, Zaid Qureshi, Jian Huang, Nam Sung Kim, Jin jun Xiong, and Wen mei Hwu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' FlatFlash: Exploiting the Byte- Accessibility of SSDs within A Unified Memory-Storage Hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of the 24th ACM International Conference on Architectural Support for Programming Languages and Operating Systems (ASPLOS’19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Providence, RI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [5] Nitin Agrawal, Vijayan Prabhakaran, Ted Wobber, John D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Davis, Mark Manasse, and Rina Panigrahy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Design Tradeoffs for SSD Performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of the USENIX 2008 Annual Technical Conference (ATC’08).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Boston, Massachusetts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [6] Yu Cai, Saugata Ghose, Erich F Haratsch, Yixin Luo, and Onur Mutlu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Error characterization, mitigation, and recovery in flash-memory-based solid-state drives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' IEEE 105, 9 (2017), 1666–1704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [7] Feng Chen, Tian Luo, and Xiaodong Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' CAFTL: A Content-Aware Flash Translation Layer Enhancing the Lifespan of Flash Memory based Solid State Drives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of the 9th USENIX Conference on File and Storage Technologies (FAST’11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' San Jose, CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [8] Renhai Chen, Zhiwei Qin, Yi Wang, Duo Liu, Zili Shao, and Yong Guan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' On- demand block-level address mapping in large-scale NAND flash storage systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 64, 6 (2014), 1729–1741.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [9] Tae-Sun Chung, Dong-Joo Park, and Jongik Kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LSTAFF*: An Efficient Flash Translation Layer for Large Block Flash Memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of the 2011 ACM Symposium on Applied Computing (SAC’11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' TaiChung Taiwan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [10] Curtis R Cook and Do Jin Kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Best sorting algorithm for nearly sorted lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' ACM 23, 11 (1980), 620–624.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [11] Yifan Dai, Yien Xu, Aishwarya Ganesan, Ramnatthan Alagappan, Brian Kroth, Andrea Arpaci-Dusseau, and Remzi Arpaci-Dusseau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' From WiscKey to Bourbon: A Learned Index for Log-Structured Merge Trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of the 14th USENIX Symposium on Operating Systems Design and Implementation (OSDI’20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Virtual Event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [12] Niv Dayan, Philippe Bonnet, and Stratos Idreos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' GeckoFTL: Scalable Flash Translation Techniques For Very Large Flash Devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of the Inter- national Conference on Management of Data (SIGMOD’16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' San Francisco, CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [13] Djellel Eddine Difallah, Andrew Pavlo, Carlo Curino, and Philippe Cudré- Mauroux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' OLTP-Bench: An Extensible Testbed for Benchmarking Relational Databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' PVLDB 7, 4 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [14] Paolo Ferragina, Fabrizio Lillo, and Giorgio Vinciguerra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Why Are Learned Indexes So Effective?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='. In Proceedings of the 37th International Conference on Machine Learning (ICML’20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Virtual Event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [15] Paolo Ferragina and Giorgio Vinciguerra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The PGM-Index: A Fully-Dynamic Compressed Learned Index with Provable Worst-Case Bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Proceedings of the VLDB Endowment 13, 8 (April 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [16] FIU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' FIU Server Traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [17] Flash Memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='org/wiki/Flash_memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [18] Fusion-io Directcache: Transparent Storage Accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='fusionio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='com/systems/directcache/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [19] Gartner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Forecast Overview: NAND Flash, Worldwide, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' https: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='gartner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='com/doc/3745121/forecast-overview-nand-flash-worldwide [20] Aayush Gupta, Youngjae Kim, and Bhuvan Urgaonkar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' DFTL: A Flash Translation Layer Employing Demand-based Selective Caching of Page-level Address Mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of the 14th International Conference on Archi- tectural Support for Programming Languages and Operating Systems (ASPLOS’09).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Washington, DC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [21] Kyuhwa Han, Hyunho Gwak, Dongkun Shin, and Joo-Young Hwang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' ZNS+: Advanced Zoned Namespace Interface for Supporting In-Storage Zone Com- paction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In 15th {USENIX} Symposium on Operating Systems Design and Imple- mentation (OSDI’21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 147–162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [22] Jian Huang, Anirudh Badam, Laura Caulfield, Suman Nath, Sudipta Sengupta, Bikash Sharma, and Moinuddin K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Qureshi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' FlashBlox: Achieving Both Performance Isolation and Uniform Lifetime for Virtualized SSDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings Jinghan Sun, Shaobo Li, Yunxin Sun, Chao Sun, Dejan Vucinic, and Jian Huang of the 15th Usenix Conference on File and Storage Technologies (FAST’17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Santa clara, CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [23] Jian Huang, Anirudh Badam, Moinuddin K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Qureshi, and Karsten Schwan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Unified Address Translation for Memory-mapped SSDs with FlashMap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Pro- ceedings of the 42nd Annual International Symposium on Computer Architecture (ISCA’15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Portland, OR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [24] Jian Huang, Daixuan Li, and Jinghan Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Learning to Drive Software- Defined Storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Workshop on Machine Learning for Systems at NIPS’22 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [25] Song Jiang, Lei Zhang, XinHao Yuan, Hao Hu, and Yu Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' S-FTL: An Efficient Address Translation for Flash Memory by Exploiting Spatial Locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of the 2011 IEEE 27th Symposium on Mass Storage Systems and Technologies (MSST’11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' IEEE Computer Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [26] William K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Josephson, Lars A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Bongo, Kai Li, and David Flynn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' DFS: A File System for Virtualized Flash Storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' on Storage 6, 3 (2010), 14:1–14:25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [27] Jun He, Sudarsun Kannan, Andrea C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Arpaci-Dusseau, Remzi H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Arpaci-Dusseau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The Unwritten Contract of Solid State Drives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of the Twelfth European Conference on Computer Systems (EuroSys’17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Belgrade, Serbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [28] Dawoon Jung, Jeong-UK Kang, Heeseung Jo, Jin-Soo Kim, and Joonwon Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Superblock FTL: A superblock-based flash translation layer with a hybrid address translation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' ACM Transactions on Embedded Computing Systems (TECS) 9, 4 (2010), 1–41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [29] Jeong-Uk Kang, Heeseung Jo, Jinsoo Kim, and Joonwon Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' A Superblock- Based Flash Translation Layer for NAND Flash Memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of the 6th International Conference on Embedded Software (EMSOFT’06).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Seoul, South Korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [30] Luyi Kang, Yuqi Xie, Weiwei Jia, Xiaohao Wang, Jongryool Kim, Changhwan Youn, Myeong Joon Kang, Jin Lim, Bruce Jacob, and Jian Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' IceClave: A Trusted Execution Environment for In-Storage Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of the 54th Annual IEEE/ACM International Symposium on Microarchitecture (MICRO’21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Virtual Event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [31] Jesung Kim, Jong Min Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Noh, Sang Lyul Min, and Yookun Cho.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' A space-efficient flash translation layer for CompactFlash systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' IEEE Transac- tions on Consumer Electronics 48, 2 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [32] Andreas Kipf, Ryan Marcus, Alexander van Renen, Mihail Stoian, Alfons Kemper, Tim Kraska, and Thomas Neumann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' RadixSpline: A Single-Pass Learned Index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of the Third International Workshop on Exploiting Artificial Intelligence Techniques for Data Management (aiDM ’20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Portland, Oregon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [33] Tim Kraska, Alex Beutel, Ed H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Chi, Jeffrey Dean, and Neoklis Polyzotis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The Case for Learned Index Structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of the 2018 International Conference on Management of Data (SIGMOD’18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Houston, TX, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [34] Hunki Kwon, Eunsam Kim, Jongmoo Choi, Donghee Lee, and Sam H Noh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Janus-FTL: Finding the optimal point on the spectrum between page and block mapping schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of the tenth ACM international conference on Embedded software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 169–178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [35] Samsung Memory Solutions Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Samsung Key Value SSD enables High Per- formance Scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='samsung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='com/semiconductor/global.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='semi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='static/ Samsung_Key_Value_SSD_enables_High_Performance_Scaling-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='pdf (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [36] Joo Hwan Lee, Hui Zhang, Veronica Lagrange, Praveen Krishnamoorthy, Xi- aodong Zhao, and Yang Seok Ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' SmartSSD: FPGA accelerated near-storage data analytics on SSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' IEEE Computer architecture letters 19, 2 (2020), 110–113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [37] Sungjin Lee, Ming Liu, Sangwoo Jun, Shuotao Xu, Jihong Kim, and Arvind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Application-managed flash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of the 14th USENIX Conference on File and Storage Technologies (FAST’16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 339–353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [38] Sungjin Lee, Dongkun Shin, Young-Jin Kim, and Jihong Kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' LAST: Locality- Aware Sector Translation for NAND Flash Memory-Based Storage Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of the SIGOPS Operating Systems Review (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [39] Sang-Won Lee, Dong-Joo Park, Tae-Sun Chung, Dong-Ho Lee, Sangwon Park, and Ha-Joo Song.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' A Log Buffer-Based Flash Translation Layer Using Fully-Associative Sector Translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' ACM Transactions on Embedded Computing Systems 6, 3 (2007), 18:1–18:27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [40] Evan Liu, Milad Hashemi, Kevin Swersky, Parthasarathy Ranganathan, and Jun- whan Ahn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' An imitation learning approach for cache replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In International Conference on Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' PMLR, 6237–6247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [41] Vikram Sharma Mailthoday, Zaid Qureshi, Weixin Liang, Ziyan Feng, Simon Gar- cia de Gonzalo, Youjie Li, Hubertus Franke, Jinjun Xiong, Jian Huang, and Wen mei Hwu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' DeepStore: In-Storage Acceleration for Intelligent Queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of the 52nd IEEE/ACM International Symposium on Microarchitecture (MICRO’19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Columbus, OH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [42] Ryan Marcus, Emily Zhang, and Tim Kraska.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' CDFShop: Exploring and Optimizing Learned Index Structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of the 2020 ACM SIGMOD International Conference on Management of Data (SIGMOD’20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Portland, OR, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='1145/3318464.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='3384706 [43] Artemiy Margaritov, Dmitri Ustiugov, Edouard Bugnion, and Boris Grot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Virtual Address Translation via Learned Page Table Indexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of the Workshop on ML for Systems at NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Montreal, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [44] Kiran Kumar Matam, Gunjae Koo, Haipeng Zha, Hung-Wei Tseng, and Murali Annavaram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' GraphSSD: Graph Semantics Aware SSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of the 46th International Symposium on Computer Architecture (ISCA’19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Phoenix, Arizona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [45] Microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' MSR Cambridge Traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [46] Jian Ouyang, Shiding Lin, Song Jiang, Yong Wang, Wei Qi, Jason Cong, and Yuanzheng Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' SDF: Software-Defined Flash for Web-Scale Internet Storage Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of 19th International Conference on Architectural Support for Programming Language and Operating Systems (ASPLOS’14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Salt Lake City, UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [47] Over 50 years of development history of Flash Memory Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='elinfor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='com/knowledge/over-50-years-of-development-history- of-flash-memory-technology-p-11271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [48] Nikolaos Papandreou, Haralampos Pozidis, Nikolas Ioannou, Thomas Parnell, Roman Pletka, Milos Stanisavljevic, Radu Stoica, Sasa Tomic, Patrick Breen, Gary Tressler, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Open block characterization and read voltage calibration of 3D QLC NAND flash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In 2020 IEEE International Reliability Physics Symposium (IRPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' IEEE, 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [49] Chanik Park, Wonmoon Cheon, Jeonguk Kang, Kangho Roh, Wonhee Cho, and Jin-Soo Kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' A reconfigurable FTL (flash translation layer) architecture for NAND flash-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' ACM Transactions on Embedded Computing Systems (TECS) 7, 4 (2008), 1–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [50] Zhiwei Qin, Yi Wang, Duo Liu, and Zili Shao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Demand-based block-level address mapping in large-scale NAND flash storage systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of the eighth IEEE/ACM/IFIP international conference on Hardware/software codesign and system synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [51] Benjamin Reidys, Peng Liu, and Jian Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' RSSD: Defend against Ran- somware with Hardware-Isolated Network-Storage Codesign and Post-Attack Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of the 27th ACM International Conference on Architec- tural Support for Programming Languages and Operating Systems (ASPLOS’22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Lausanne, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [52] Benjamin Reidys, Jinghan Sun, Anirudh Badam, Shadi Noghabi, and Jian Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' BlockFlex: Enabling Storage Harvesting with Software-Defined Flash in Modern Cloud Platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of the 16th USENIX Symposium on Operating Systems Design and Implementation (OSDI’22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Carlsbad, CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [53] Liana V Rodriguez, Farzana Yusuf, Steven Lyons, Eysler Paz, Raju Rangaswami, Jason Liu, Ming Zhao, and Giri Narasimhan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Learning Cache Replacement with CACHEUS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In 19th USENIX Conference on File and Storage Technologies (FAST’21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 341–354.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [54] Zhenyuan Ruan, Tong He, and Jason Cong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' INSIDER: Designing In-Storage Computing System for Emerging High-Performance Drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of the 2019 USENIX Annual Technical Conference (USENIX ATC’19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Renton, WA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [55] Samsung Z-NAND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='samsung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='com/semiconductor/ssd/z-ssd/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [56] Subhash Sethumurugan, Jieming Yin, and John Sartori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Designing a Cost- Effective Cache Replacement Policy using Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In 2021 IEEE Inter- national Symposium on High-Performance Computer Architecture (HPCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' IEEE, 291–303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [57] Zhan Shi, Xiangru Huang, Akanksha Jain, and Calvin Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Applying deep learning to the cache replacement problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of the 52nd Annual IEEE/ACM International Symposium on Microarchitecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 413–425.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [58] smartssd 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' SmartSSD Computational Storage Drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='xilinx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='com/ applications/data-center/computational-storage/smartssd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [59] Vasily Tarasov, Erez Zadok, and Spencer Shepler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Filebench: A flexible framework for file system benchmarking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The USENIX Magazine 41, 1 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [60] Usman Saleem, Advanced SSD Buying Guide - NAND Types, DRAM Cache, HMB Explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' https://appuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='com/ssd-buying-guide/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [61] Dana Van Aken, Djellel E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Difallah, Andrew Pavlo, Carlo Curino, and Philippe Cudré-Mauroux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' BenchPress: Dynamic Workload Control in the OLTP- Bench Testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of the 2015 ACM SIGMOD International Conference on Management of Data (SIGMOD’15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [62] Xiaohao Wang, Yifan Yuan, You Zhou, Chance C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Coats, and Jian Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Project Almanac: A Time-Traveling Solid-State Drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of the 14th European Conference on Computer Systems (EuroSys’19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Dresden, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [63] Nan Wu and Yuan Xie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' A Survey of Machine Learning for Computer Architecture and Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' CoRR abs/2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='07952 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='org/abs/ 2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content='07952 [64] Qing Xie, Chaoyi Pang, Xiaofang Zhou, Xiangliang Zhang, and Ke Deng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Maximum Error-Bounded Piecewise Linear Representation for Online Stream Approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Proceedings of the VLDB Journal 23, 6 (Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [65] Qing Xie, Chaoyi Pang, Xiaofang Zhou, Xiangliang Zhang, and Ke Deng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Maximum error-bounded piecewise linear representation for online stream ap- proximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' The VLDB journal 23, 6 (2014), 915–937.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' [66] Yiying Zhang, Leo Prasath Arulraj, Andrea C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Arpaci-Dusseau, and Remzi H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' Arpaci-Dusseau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' De-indirection for Flash-based SSDs with Nameless Writes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' In Proceedings of the 10th USENIX Conference on File and Storage Technologies (FAST’12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} +page_content=' San Jose, CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfRvd3/content/2301.00072v1.pdf'} diff --git a/AdE2T4oBgHgl3EQfRQcf/content/tmp_files/2301.03778v1.pdf.txt b/AdE2T4oBgHgl3EQfRQcf/content/tmp_files/2301.03778v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b31ad077232401823ddfc134809cf33db35618c4 --- /dev/null +++ b/AdE2T4oBgHgl3EQfRQcf/content/tmp_files/2301.03778v1.pdf.txt @@ -0,0 +1,833 @@ +arXiv:2301.03778v1 [quant-ph] 10 Jan 2023 +Letter +Optics Letters +1 +Efficient and robust chiral discrimination by +invariant-based inverse engineering +HANG XU1, XUE-KE SONG1,2, LIU YE1, AND DONG WANG1,3 +1School of Physics and Optoelectronics Engineering, Anhui University, Hefei 230601, China +2Corresponding author: songxk@ahu.edu.cn +3Corresponding author: dwang@ahu.edu.cn +Compiled January 11, 2023 +We propose an accurate and convenient method to +achieve 100% discrimination of chiral molecules with +Lewis-Riesenfeld invariant. By reversely designing the +pulse scheme of handed resolution, we obtain the pa- +rameters of the three-level Hamiltonians to achieve this +goal. For the same initial state, we can completely trans- +fer its population to one energy level for left-handed +molecules, while transfer it to another energy level for +right-handed molecules. +Moreover, this method can +be further optimized when errors exist, and it shows +that the optimal method are more robust against these +errors than the counterdiabatic and original invariant- +based shortcut schemes. This provides an effective, ac- +curate, and robust method to distinguish the handed- +ness of molecules. +© 2023 Optica Publishing Group +http://dx.doi.org/10.1364/ao.XX.XXXXXX +Chirality, which was first proposed by Pasteur in 1848 [1] +originating from symmetry breaking [2], is a very important +concept or attribute in natural science. It has attracted exten- +sive attentions in specific fields of physics, materials science +[3], chemistry [4], biology [5], and medicine [6]. In principle, +when the atomic distribution and chemical bond structure of +two molecules are symmetrical in the mirror image but cannot +coincide, these molecules possess chirality with left (L) hand- +edness or right (R) handedness. Generally, molecules with dif- +ferent chirality show the same physical and chemical proper- +ties. However, in some specific cases, they show dramatically +opposite properties, especially biological activity [7]. The drug +molecules must match the geometric structure of the receptor +(reactive substance) molecules in order to have the proper effi- +cacy. +In recent years, there are many studies [8–18] to use quan- +tum coherent manipulation techniques to realize the effective +discrimination of chiral molecules, including adiabatic pas- +sages [19], counter-diabatic driving [20–23], composite pulses +[24–26], etc. In 2019, Vitanov et al. [9] proposed an efficient +chiral resolution using delayed pulses based on the principle of +counter-diabatic quantum driving. In 2019, Ye et al. [10] showed +two dynamic methods to achieve inner-state enantioseparation +in the case that the handedness system is reduced to a effec- +tive two-level system. In 2020, Torosov et al. [11] introduced +a method for the chiral molecule detection using sequences of +three pulses, and the composite pulses are used to realize the +robustness to the area error. +In this paper, we propose an efficient and robust chiral res- +olution method based on optimal Lewis-Riesenfeld invariant +(LRI) shortcut. +For the three-level Hamiltonians of the left- +handed and right-handed molecules, we can design the invari- +ants of the corresponding L and R systems [27–32], respectively. +The systems are evolved along eigenstates of their respective +invariants from the same initial energy level, while they will +reach to different final energy levels with regard to different +chiral molecules. This means that a 100% chiral resolution is +achieved. The advantage of LRI is that it has a large parameter +selections to be further optimized with respect to various con- +trol errors. Taking systematic and detuning errors into account, +we find that the optimal invariant shortcut scheme are more ro- +bust against these errors compared to the counter-diabatic and +the original invariant shortcuts. +Let us consider a typical cyclic three-level system [33], as +shown in Fig. 1. The Hamiltonian, in the bases {|1⟩ , |2⟩ , |3⟩}, +reads +HL,R +0 += ¯h + + + + + +0 +Ωp +∓Ωqeiγ +Ωp +0 +Ωs +∓Ωqe−iγ +Ωs +0 + + + + + , +(1) +where the superscripts L and R denote the left-handedness and +right-handedness. Ωp, Ωs , and Ωq represent the Rabi frequen- +cies of the three energy level transitions, respectively. The sign +− or + of Ωq represents L or R handedness. γ is the phase of +Ωq, in this paper, we set γ = π/2 and Ωp = Ωs = Ω. Therefore, +the simplified Hamiltonian is +HL,R = ¯h + + + + + +0 +Ω +∓iΩq +Ω +0 +Ω +±iΩq +Ω +0 + + + + + , +(2) +In order to achieve accurate chiral resolution, the goal is that +after applying the same specific pulse to the two chiral systems, +the final state of the left-handedness system is completely at one +energy level, and the final state of the right-handedness system + +Letter +Optics Letters +2 + +! +!! +"!" +!" +!! +! + +# +# +$ +$ +%&' +%(' +Fig. 1. Schematic diagram of chiral molecules with L (a) and +R (b) handedness in three different energy levels. Their dipole +transitions are mirror symmetric, with the same Ωp and Ωs +but the Ωq with opposite sign. +is completely at another energy level, so that we can determine +its chirality by measuring the energy spectrum of the system. +First, we consider the L chiral system. The invariant is +IL = + + + + + +0 +sin ϕ sin θ +−i cos ϕ +sin ϕ sin θ +0 +sin ϕ cos θ +i cos ϕ +sin ϕ cos θ +0 + + + + + . +(3) +The eigenstates of the invariant are +��φL +0 +� = + + + + + +− sin ϕ cos θ +i cos ϕ +sin ϕsinθ + + + + + , +��φL± +� = +1 +√ +2 + + + + + +cos ϕ cos θ ± i sin θ +i sin ϕ +− cos ϕ sin θ ± i cos θ + + + + + , +(4) +with corresponding eigenvalues µ0 = 0 and µ± = ±1. By solv- +ing the dynamical equation [31], the following constraint condi- +tions are obtained: +Ω = ˙ϕ/(sin θ − cos θ), +Ωq = ˙ϕ cot ϕ(sin θ + cos θ)/(sin θ − cos θ) − ˙θ, +(5) +where the dot represents the derivative with respect to time. +When the above conditions are satisfied, we can write the gen- +eral solution +��ψL(t) +� +of Schrödinger [27] as +���ψL(t) +� += ∑ +j=0,± +Bjeiηj(t) ���φL +j (t) +� +, +(6) +where Bj are time-independent constants, and ηj(t) are the so- +called LR phases which satisfy +˙ηj(t) = 1 +¯h +� +φL +j (t) +��� i¯h ∂ +∂t − HL ���φL +j (t) +� +. +(7) +Thereby, we can get +η0(t) = 0, +η±(t) = ± � t +0 dt′ [ ˙ϕ csc ϕ(sin θ + cos θ)/(cos θ − sin θ)]. +(8) +0 +0.25 +0.5 +0.75 +1 +t (T) +0 +0.5 +1 +Populations of L +0 +0.25 +0.5 +0.75 +1 +t (T) +0 +0.5 +1 +Populations of R +(b) +(a) +Fig. 2. Schematic diagram of energy level populations of the +L(R) systems using SPS. (a) Populations vs the time t of L sys- +tem; (b) Populations vs the time t of R system. Red dashed, +green solid, blue dotted lines stand for the populations of |1⟩, +|2⟩, and |3⟩, respectively. +It can be seen from the Eq. (6) that if the L-handed system is +initially in an eigenstate +���φL +j (t) +� +, it will also be in this eigenstate +at any time after time evolution. As for the eigenstate +��φL +0 (t) +� +, +if the boundary conditions of the parameters are chosen as +ϕ(0) = 0, +ϕ(T) = π/2, +θ(T) = π/2, +(9) +where T is final time moment, the L system will completely +transfer to the level |3⟩ if initially in the level |2⟩. Second, let +us consider the R system. We set its invariant as +IR = + + + + + +0 +sin ϕ cos θ +i cos ϕ +sin ϕ cos θ +0 +sin ϕ sin θ +−i cos ϕ +sin ϕ sin θ +0 + + + + + . +(10) +Similarly, we can obtain the eigenstates of this invariant IR: +��φR +0 +� = + + + + + +sin ϕsinθ +i cos ϕ +− sin ϕ cos θ + + + + + , +��φR± +� = +1 +√ +2 + + + + + +− cos ϕ sin θ ± i cos θ +i sin ϕ +cos ϕ cos θ ± i sin θ + + + + + , +(11) +with corresponding eigenvalues µ0 = 0 and µ± = ±1. For the +R system, we find that the parameter constraints and LR phases +of R system are exactly the same as those of L system, as shown +in Eq. (5) and (8). This means that if we drive the L or R sys- +tem to evolve along the eigenstate +��φL +0 +� +or +��φR +0 +� +, we can apply +the same pulse scheme by inversely solving the constraint con- +ditions in Eq. (5). +Now, we pay attention to the eigenstate +��φR +0 (t) +� +. If we have +the same boundary condition in Eq. (9), the R system will com- +pletely transfer from |2⟩ to |1⟩ for t ∈ [0, T], which is completely +different from the target energy level of the L system. That is to +say, we can apply the same pulse to a pair of L and R systems +when they are initially at the level |2⟩ by choosing the invariant +parameters to fulfill the boundary condition in Eq. (9). This can +drive the L system to fully evolve to the level |3⟩, while drive +the R system to fully evolve to the level |1⟩. Finally, their hand- +edness can be determined by measuring their energy spectrum. +As a result, the 100% chiral discrimination is reached. + +Letter +Optics Letters +3 +-0.2 +-0.1 +0 +0.1 +0.2 +error amplitude α +0.9 +0.95 +1 +Fidelity +Fig. 3. The systematic error amplitude α vs fidelity of different +schemes: SPS (blue, dotted line), OSE (red, dashed line), and +CD (green, solid line). +Here, we consider a simple parameter scheme (SPS) to show +how to achieve an efficient chiral discrimination by invariant- +based inverse engineering. When we choose +ϕ(t) = πt +2T , +θ(t) = π +2 , +(12) +to satisfy the boundary conditions in Eq. (9). Inversely, we can +get the parameters of Hamiltonian, from the Eq. (5), as +Ω = π +2T , +Ωq = π +2 cot πt +2T , +(13) +where T is pulse duration and t ∈ [0, T]. In Fig. 2, we plot +the evolution curve of the level population of the L and R sys- +tems. It can be seen that the two systems are initially at the +same level |2⟩. At t = T, the population of the L system com- +pletely transfers to level |3⟩, while the population of the R sys- +tem completely transfers to level |1⟩. Therefore, through mea- +suring their energy spectrum or population of the system, we +can determine its chirality: if the population of the state |3⟩ is +1, this is a left-handed system, and if the population of the state +|1⟩ is 1, it is a right-handed system. +On the other hand, when we consider the influence of con- +trol errors that may occur in the experiment on the fidelity (or +discrimination) of the resolution scheme, it is necessary to op- +timize the LRI scheme with respect to these errors. We use +a new Hamiltonian H′ to indicate the existence of errors, i.e., +H → H′ = H + He, where He is error Hamiltonian. The fidelity +is generally defined as +F = +��� +ψ(T) +�� ψ′(T) +���2, +(14) +where |ψ(T)⟩ is target state and |ψ′(T)⟩ is actual state of system +at the final moment T. Using perturbation theory [30], we have +FL,R ≈ 1 − 1 +¯h2 ∑ +± +���� +� T +0 dt +� +φL,R +0 +(t) +��� He +���φL,R +j +(t) +� +eiηj(t) +���� +2 +. +(15) +We first consider the influence of systematic error. In this case, +the error Hamiltonian is described as +HL,R +e += αHL,R, +(16) +-1 +-0.5 +0 +0.5 +1 +error amplitude δ (1/T) +0.9 +0.95 +1 +Fidelity +Fig. 4. The detuning error amplitude δ vs fidelity of different +schemes: SPS (blue, dotted line), OSD (red, dashed line), and +CD (green, solid line). +where, α is a dimensionless parameter, representing the ampli- +tude of systematic error. Combining Eqs. (15) and (16), we can +get +FL = FR = 1 − α2 +���� +� T +0 ( ˙θ sin ϕ + i ˙ϕ)eiη+(t)dt +���� +2 +. +(17) +Obviously, the fidelity of the target level for the L and R sys- +tems is affected by the systematic error in the same way. There- +fore, we only need to analyze the influence of error on the fi- +delity of the L or R system. The systematic error sensitivity is +defined as +qα = − ∂2FL,R +2∂α2 |α=0 = − ∂FL,R +∂(α2) |α=0. +(18) +The smaller the sensitivity, the smaller the impact of error on +fidelity. Then we have +qα = +���� +� T +0 ( ˙θ sin ϕ + i ˙ϕ)eiη+(t)dt +���� +2 +. +(19) +To meet the boundary conditions, we still choose +ϕ(t) = πt +2T . +(20) +We do not set the form of θ(t) at first, but try the Fourier series +type of Ansatz with regard to the LR phase η+ +η+(t) = −[n sin(3ϕ) − ϕ], +(21) +where n is a real number that can be chosen freely. From the +Eq. (8), the parameter θ(t) takes the form +θ(t) = arccot3n cos(3ϕ) sin ϕ − sin ϕ + 1 +3n cos(3ϕ) sin ϕ − sin ϕ − 1, +(22) +which satisfies the boundary condition θ(T) = π/2. +Based +on the above equations, we can calculate the systematic error +sensitivity qα numerically. When n = 1.07, the systematic er- +ror sensitivity reaches the minimum value of 0.52, which is +defined as the optimal scheme for systematic error sensitiv- +ity (OSS). In Fig. 3, we compare the influence of systematic +error on the fidelity or discrimination with several coherent + +Letter +Optics Letters +4 +Fig. 5. Fidelity FL,R vs the systematic error amplitude α and +detuning error amplitude δ by LRI scheme of n = 1.10. The +yellow area in the middle corresponds to FL,R ≥ 0.99. +control schemes, including OSS, SPS, and the counter-dabatic +(CD) shortcut method in Ref. [9]. We can observe that all these +schemes can achieve 100% discrimination in the absence of the +error, and the OSE scheme is the most robust against systematic +error, followed by SPS, and finally CD. +Another important error in experiment is the detuning error. +In this case, the error Hamiltonian is +HL,R +e += δ¯h(|3⟩ ⟨3| − |1⟩ ⟨1|), +(23) +where δ represents the detuning amplitude, and its unit is 1/T. +In the same way, we can obtain the fidelity as +FL = FR += 1 − δ2 +4 +��� +� T +0 [cos(2θ) sin(2ϕ) + 2i sin(2θ) sin ϕ]eiη+(t)dt +��� +2 +. +(24) +And we have +qδ = +���� +� T +0 [cos(2θ) sin(2ϕ) + 2i sin(2θ) sin ϕ]eiη+(t)dt +���� +2 +. +(25) +Here, the parameters ϕ and η+ are chosen as the same forms +of Eqs. (20) and (21). We can find that, the detuning error sen- +sitivity reaches the minimum value 0 when n = 1.12. We call +the corresponding parameter scheme as the optimal scheme for +detuning error (OSD). In Fig. 4, we compare the influence of +detuning error on the fidelity or discrimination with OSD, SPS, +and CD control schemes. Again, the OSD scheme is the most +robust against the detuning error. Furthermore, we plot how +the fidelity is affected by the systematic error and detuning er- +ror in Fig. 5. It can be seen that the optimal scheme shows high +robustness against these two errors with a broad range of high +efficiencies over 99% . +In conclusion, we propose a highly efficient and robust chiral +discrimination method for the cyclic three-level systems of chi- +ral molecules based on the invariant-based inverse engineering. +Through applying to the same pulse on the three-level system, +molecules with different chirality will transit to different energy +levels. The L system stay in |3⟩ and R system stay in |1⟩ at the +final time from the same initial state. We can realize the 100% +chiral discrimination of molecules by measuring population or +energy spectrum. Moreover, we can design the corresponding +optimization schemes with respect to different experimental er- +rors. By comparison, the optimization schemes are superior to +the SPS and the CD schemes. +Funding. +This study was supported by the National Natural Sci- +ence Foundation of China (Grant No. 12004006, No. 12075001, and +No. 12175001), Anhui Provincial Key Research and Development Plan +(Grant No. 2022b13020004), and the Anhui Provincial Natural Science +Foundation (Grant No. 2008085QA43). +Disclosures. +The authors declare no conflicts of interest. +Data Availability Statement. +Data underlying the results pre- +sented in this Letter are not publicly available at this time but may be +obtained from the authors upon reasonable request. +REFERENCES +1. +L. Pasteur, Ann. Chim. Phys. 24, 442 (1848). +2. +R. F. Dashen, Phys. Rev. D 3, 1879 (1971). +3. +M. Fu, F. Liu, and L. Hu, Compos. Sci. Technol. 160, 111 (2018). +4. +Z.-G. Gu, C. Zhan, J. Zhang, and X. Bu, Chem. Soc. Rev. 45, 3122 +(2016). +5. +T. J. Leitereg, D. G. Guadagni, J. Harris, T. R. Mon, and R. Teranishi, +J. Agric. Food Chem. 19, 785 (1971). +6. +A. J. Hutt and S. C. Tan, Drugs 52, 1 (1996). +7. +J. Gal, Chirality 12, 959 (2012). +8. +M. Shapiro, E. Frishman, and P. Brumer, Phys. Rev. Lett. 84, 1669 +(2000). +9. +N. V. Vitanov and M. Drewsen, Phys. Rev. Lett. 122, 173202 (2019). +10. +C. Ye, Q. Zhang, Y.-Y. Chen, and Y. Li, Phys. Rev. A 100, 043403 +(2019). +11. +B. T. Torosov, M. Drewsen, and N. V. Vitanov, Phys. Rev. A 101, +063401 (2020). +12. +B. T. Torosov, M. Drewsen, and N. V. Vitanov, Phys. Rev. Res. 2, +043235 (2020). +13. +J.-L. Wu, Y. Wang, J. Song, Y. Xia, S.-L. Su, and Y.-Y. Jiang, Phys. +Rev. A 100, 043413 (2019). +14. +J.-L. Wu, S.-L. Su, Y. Xia, Y.-Y. Jiang, and J. Song, Opt. Express 28, +33475 (2020). +15. +J.-L. Wu, Y. Wang, J.-X. Han, C. Wang, S.-L. Su, Y. Xia, Y.-Y. Jiang, +and J. Song, Phys. Rev. Appl. 13, 044021 (2020). +16. +Y.-H. Kang, Z.-C. Shi, J. Song, and Y. Xia, Opt. Lett. 45, 4952 (2020). +17. +C. Ye, Q.-S. Zhang, Y.-Y. Chen, and Y. Li, Phys. Rev. Res. 2, 033064 +(2020). +18. +Y. Guo, X. Gong, S. Ma, and C.-C. Shu, Phys. Rev. A 105, 013102 +(2022). +19. +N. V. Vitanov, L. P. Yatsenko, and K. Bergmann, Phys. Rev. A 68, +043401 (2003). +20. +M. Demirplak and S. A. Rice, J. Phys. Chem. A 107, 9937 (2003). +21. +M. V. Berry, J. Phys. A 42, 365303 (2009). +22. +X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, +Phys. Rev. Lett. 105, 123003 (2010). +23. +X.-K. Song, Q. Ai, J. Qiu, and F.-G. Deng, Phys. Rev. A 93, 052324 +(2016). +24. +B. T. Torosov, S. Guérin, and N. V. Vitanov, Phys. Rev. Lett. 106, +233001 (2011). +25. +G. T. Genov, D. Schraft, T. Halfmann, and N. V. Vitanov, Phys. Rev. +Lett. 113, 043001 (2014). +26. +G. T. Genov, D. Schraft, N. V. Vitanov, and T. Halfmann, Phys. Rev. +Lett. 118, 133202 (2017). +27. +H. R. Lewis and W. B. Riesenfeld, J. Math. Phys. 10, 1458 (1969). +28. +X. Chen, A. Ruschhaupt, S. Schmidt, A. del Campo, D. Guéry-Odelin, +and J. G. Muga, Phys. Rev. Lett. 104, 063002 (2010). +29. +X. Chen and J. G. Muga, Phys. Rev. A 86, 033405 (2012). +30. +A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, New J. Phys. 14, +093040 (2012). +31. +S.-F. Qi and J. Jing, Phys. Rev. A 105, 053710 (2022). +32. +Y.-H. Kang, Y.-H. Chen, X. Wang, J. Song, Y. Xia, A. Miranowicz, S.-B. +Zheng, and F. Nori, Phys. Rev. Res. 4, 013233 (2022). +33. +R. Unanyan, L. Yatsenko, K. Bergmann, and B. Shore, Opt. Commun. +139, 48 (1997). + ++2 +ude0.950.9error +0 +b00.85etunin0.8 +20 +litude0 +amp +rrortematic-2 +-0.2 +sysLetter +Optics Letters +5 +FULL REFERENCES +1. +L. Pasteur, “On the Relations Crystalline Form, Chemical Composition +and Direction of Polarization Rotatorie,” Ann. Chim. Phys. 24, 442–459 +(1848). +2. +R. F. Dashen, “Some features of chiral symmetry breaking,” Phys. Rev. +D 3, 1879 (1971). +3. +M. Fu, F. Liu, and L. Hu, “A novel category of 3d chiral material with +negative Poisson’s ratio,” Compos. Sci. Technol. 160, 111–118 (2018). +4. +Z.-G. Gu, C. Zhan, J. Zhang, and X. Bu, “Chiral chemistry of +metal–camphorate frameworks,” Chem. Soc. Rev. 45, 3122–3144 +(2016). +5. +T. J. Leitereg, D. G. Guadagni, J. Harris, T. R. Mon, and R. Teranishi, +“Chemical and sensory data supporting the difference between the +odors of the enantiomeric carvones,” J. Agric. Food Chem. 19, 785– +787 (1971). +6. +A. J. Hutt and S. C. Tan, “Drug chirality and its clinical significance,” +Drugs 52, 1–12 (1996). +7. +J. Gal, “The discovery of stereoselectivity at biological receptors: Ar- +naldo Piutti and the taste of the asparagine enantiomers—History and +analysis on the 125th anniversary,” Chirality 12, 959–976 (2012). +8. +M. Shapiro, E. Frishman, and P. Brumer, “Coherently controlled asym- +metric synthesis with achiral light,” Phys. Rev. Lett. 84, 1669 (2000). +9. +N. V. Vitanov and M. Drewsen, “Highly efficient detection and separa- +tion of chiral molecules through shortcuts to adiabaticity,” Phys. Rev. +Lett. 122, 173202 (2019). +10. +C. Ye, Q. Zhang, Y.-Y. Chen, and Y. Li, “Effective two-level models +for highly efficient inner-state enantioseparation based on cyclic three- +level systems of chiral molecules,” Phys. Rev. A 100, 043403 (2019). +11. +B. T. Torosov, M. Drewsen, and N. V. Vitanov, “Efficient and robust chi- +ral resolution by composite pulses,” Phys. Rev. A 101, 063401 (2020). +12. +B. T. Torosov, M. Drewsen, and N. V. Vitanov, “Chiral resolution by +composite Raman pulses,” Phys. Rev. Res. 2, 043235 (2020). +13. +J.-L. Wu, Y. Wang, J. Song, Y. Xia, S.-L. Su, and Y.-Y. Jiang, “Robust +and highly efficient discrimination of chiral molecules through three- +mode parallel paths,” Phys. Rev. A 100, 043413 (2019). +14. +J.-L. Wu, S.-L. Su, Y. Xia, Y.-Y. Jiang, and J. Song, “Discrimination of +enantiomers through quantum interference and quantum Zeno effect,” +Opt. Express 28, 33475–33489 (2020). +15. +J.-L. Wu, Y. Wang, J.-X. Han, C. Wang, S.-L. Su, Y. Xia, Y.-Y. Jiang, +and J. Song, “Two-Path Interference for Enantiomer-Selective State +Transfer of Chiral Molecules,” Phys. Rev. Appl. 13, 044021 (2020). +16. +Y.-H. Kang, Z.-C. Shi, J. Song, and Y. Xia, “Effective discrimination of +chiral molecules in a cavity,” Opt. Lett. 45, 4952–4955 (2020). +17. +C. Ye, Q.-S. Zhang, Y.-Y. Chen, and Y. Li, “Fast enantioconversion of +chiral mixtures based on a four-level double-∆ model,” Phys. Rev. Res. +2, 033064 (2020). +18. +Y. Guo, X. Gong, S. Ma, and C.-C. Shu, “Cyclic three-level-pulse-area +theorem for enantioselective state transfer of chiral molecules,” Phys. +Rev. A 105, 013102 (2022). +19. +N. V. Vitanov, L. P. Yatsenko, and K. Bergmann, “Population transfer +by an amplitude-modulated pulse,” Phys. Rev. A 68, 043401 (2003). +20. +M. Demirplak and S. A. Rice, “Adiabatic population transfer with con- +trol fields,” J. Phys. Chem. A 107, 9937–9945 (2003). +21. +M. V. Berry, “Transitionless quantum driving,” J. Phys. A 42, 365303 +(2009). +22. +X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, +“Shortcut to Adiabatic Passage in Two- and Three-Level Atoms,” Phys. +Rev. Lett. 105, 123003 (2010). +23. +X.-K. Song, Q. Ai, J. Qiu, and F.-G. Deng, “Physically feasible three- +level transitionless quantum driving with multiple Schrödinger dynam- +ics,” Phys. Rev. A 93, 052324 (2016). +24. +B. T. Torosov, S. Guérin, and N. V. Vitanov, “High-Fidelity Adiabatic +Passage by Composite Sequences of Chirped Pulses,” Phys. Rev. +Lett. 106, 233001 (2011). +25. +G. T. Genov, D. Schraft, T. Halfmann, and N. V. Vitanov, “Correction of +Arbitrary Field Errors in Population Inversion of Quantum Systems by +Universal Composite Pulses,” Phys. Rev. Lett. 113, 043001 (2014). +26. +G. T. Genov, D. Schraft, N. V. Vitanov, and T. Halfmann, “Arbitrarily +Accurate Pulse Sequences for Robust Dynamical Decoupling,” Phys. +Rev. Lett. 118, 133202 (2017). +27. +H. R. Lewis and W. B. Riesenfeld, “An exact quantum theory of the +time-dependent harmonic oscillator and of a charged particle in a +time-dependent electromagnetic field,” J. Math. Phys. 10, 1458–1473 +(1969). +28. +X. Chen, A. Ruschhaupt, S. Schmidt, A. del Campo, D. Guéry-Odelin, +and J. G. Muga, “Fast Optimal Frictionless Atom Cooling in Harmonic +Traps: Shortcut to Adiabaticity,” Phys. Rev. Lett. 104, 063002 (2010). +29. +X. Chen and J. G. Muga, “Engineering of fast population transfer in +three-level systems,” Phys. Rev. A 86, 033405 (2012). +30. +A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, “Optimally robust +shortcuts to population inversion in two-level quantum systems,” New +J. Phys. 14, 093040 (2012). +31. +S.-F. Qi and J. Jing, “Accelerated adiabatic passage in cavity mag- +nomechanics,” Phys. Rev. A 105, 053710 (2022). +32. +Y.-H. Kang, Y.-H. Chen, X. Wang, J. Song, Y. Xia, A. Miranowicz, S.- +B. Zheng, and F. Nori, “Nonadiabatic geometric quantum computation +with cat-state qubits via invariant-based reverse engineering,” Phys. +Rev. Res. 4, 013233 (2022). +33. +R. Unanyan, L. Yatsenko, K. Bergmann, and B. Shore, “Laser-induced +adiabatic atomic reorientation with control of diabatic losses,” Opt. +Commun. 139, 48–54 (1997). + diff --git a/AdE2T4oBgHgl3EQfRQcf/content/tmp_files/load_file.txt b/AdE2T4oBgHgl3EQfRQcf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7f75274ca723cbfc415be26aa0b60495b59c7aa1 --- /dev/null +++ b/AdE2T4oBgHgl3EQfRQcf/content/tmp_files/load_file.txt @@ -0,0 +1,810 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf,len=809 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='03778v1 [quant-ph] 10 Jan 2023 Letter Optics Letters 1 Efficient and robust chiral discrimination by invariant-based inverse engineering HANG XU1, XUE-KE SONG1,2, LIU YE1, AND DONG WANG1,3 1School of Physics and Optoelectronics Engineering, Anhui University, Hefei 230601, China 2Corresponding author: songxk@ahu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='cn 3Corresponding author: dwang@ahu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='cn Compiled January 11, 2023 We propose an accurate and convenient method to achieve 100% discrimination of chiral molecules with Lewis-Riesenfeld invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' By reversely designing the pulse scheme of handed resolution, we obtain the pa- rameters of the three-level Hamiltonians to achieve this goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' For the same initial state, we can completely trans- fer its population to one energy level for left-handed molecules, while transfer it to another energy level for right-handed molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Moreover, this method can be further optimized when errors exist, and it shows that the optimal method are more robust against these errors than the counterdiabatic and original invariant- based shortcut schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' This provides an effective, ac- curate, and robust method to distinguish the handed- ness of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' © 2023 Optica Publishing Group http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='1364/ao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='XX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='XXXXXX Chirality, which was first proposed by Pasteur in 1848 [1] originating from symmetry breaking [2], is a very important concept or attribute in natural science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' It has attracted exten- sive attentions in specific fields of physics, materials science [3], chemistry [4], biology [5], and medicine [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' In principle, when the atomic distribution and chemical bond structure of two molecules are symmetrical in the mirror image but cannot coincide, these molecules possess chirality with left (L) hand- edness or right (R) handedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Generally, molecules with dif- ferent chirality show the same physical and chemical proper- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' However, in some specific cases, they show dramatically opposite properties, especially biological activity [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' The drug molecules must match the geometric structure of the receptor (reactive substance) molecules in order to have the proper effi- cacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' In recent years, there are many studies [8–18] to use quan- tum coherent manipulation techniques to realize the effective discrimination of chiral molecules, including adiabatic pas- sages [19], counter-diabatic driving [20–23], composite pulses [24–26], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' In 2019, Vitanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' [9] proposed an efficient chiral resolution using delayed pulses based on the principle of counter-diabatic quantum driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' In 2019, Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' [10] showed two dynamic methods to achieve inner-state enantioseparation in the case that the handedness system is reduced to a effec- tive two-level system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' In 2020, Torosov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' [11] introduced a method for the chiral molecule detection using sequences of three pulses, and the composite pulses are used to realize the robustness to the area error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' In this paper, we propose an efficient and robust chiral res- olution method based on optimal Lewis-Riesenfeld invariant (LRI) shortcut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' For the three-level Hamiltonians of the left- handed and right-handed molecules, we can design the invari- ants of the corresponding L and R systems [27–32], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' The systems are evolved along eigenstates of their respective invariants from the same initial energy level, while they will reach to different final energy levels with regard to different chiral molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' This means that a 100% chiral resolution is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' The advantage of LRI is that it has a large parameter selections to be further optimized with respect to various con- trol errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Taking systematic and detuning errors into account, we find that the optimal invariant shortcut scheme are more ro- bust against these errors compared to the counter-diabatic and the original invariant shortcuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Let us consider a typical cyclic three-level system [33], as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' The Hamiltonian, in the bases {|1⟩ , |2⟩ , |3⟩}, reads HL,R 0 = ¯h \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed 0 Ωp ∓Ωqeiγ Ωp 0 Ωs ∓Ωqe−iγ Ωs 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , (1) where the superscripts L and R denote the left-handedness and right-handedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Ωp, Ωs , and Ωq represent the Rabi frequen- cies of the three energy level transitions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' The sign − or + of Ωq represents L or R handedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' γ is the phase of Ωq, in this paper, we set γ = π/2 and Ωp = Ωs = Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Therefore, the simplified Hamiltonian is HL,R = ¯h \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed 0 Ω ∓iΩq Ω 0 Ω ±iΩq Ω 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , (2) In order to achieve accurate chiral resolution, the goal is that after applying the same specific pulse to the two chiral systems, the final state of the left-handedness system is completely at one energy level, and the final state of the right-handedness system Letter Optics Letters 2 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' "!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='" !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='" !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=" # # $ $ %&' %(' Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Schematic diagram of chiral molecules with L (a) and R (b) handedness in three different energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Their dipole transitions are mirror symmetric, with the same Ωp and Ωs but the Ωq with opposite sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' is completely at another energy level, so that we can determine its chirality by measuring the energy spectrum of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' First, we consider the L chiral system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' The invariant is IL = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed 0 sin ϕ sin θ −i cos ϕ sin ϕ sin θ 0 sin ϕ cos θ i cos ϕ sin ϕ cos θ 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' (3) The eigenstates of the invariant are ��φL 0 � = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed − sin ϕ cos θ i cos ϕ sin ϕsinθ \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , ��φL± � = 1 √ 2 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed cos ϕ cos θ ± i sin θ i sin ϕ − cos ϕ sin θ ± i cos θ \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , (4) with corresponding eigenvalues µ0 = 0 and µ± = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' By solv- ing the dynamical equation [31], the following constraint condi- tions are obtained: Ω = ˙ϕ/(sin θ − cos θ), Ωq = ˙ϕ cot ϕ(sin θ + cos θ)/(sin θ − cos θ) − ˙θ, (5) where the dot represents the derivative with respect to time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' When the above conditions are satisfied, we can write the gen- eral solution ��ψL(t) � of Schrödinger [27] as ���ψL(t) � = ∑ j=0,± Bjeiηj(t) ���φL j (t) � , (6) where Bj are time-independent constants, and ηj(t) are the so- called LR phases which satisfy ˙ηj(t) = 1 ¯h � φL j (t) ��� i¯h ∂ ∂t − HL ���φL j (t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' (7) Thereby, we can get η0(t) = 0, η±(t) = ± � t 0 dt′ [ ˙ϕ csc ϕ(sin θ + cos θ)/(cos θ − sin θ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' (8) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='75 1 t (T) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='5 1 Populations of L 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='75 1 t (T) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='5 1 Populations of R (b) (a) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Schematic diagram of energy level populations of the L(R) systems using SPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' (a) Populations vs the time t of L sys- tem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' (b) Populations vs the time t of R system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Red dashed, green solid, blue dotted lines stand for the populations of |1⟩, |2⟩, and |3⟩, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' It can be seen from the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' (6) that if the L-handed system is initially in an eigenstate ���φL j (t) � , it will also be in this eigenstate at any time after time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' As for the eigenstate ��φL 0 (t) � , if the boundary conditions of the parameters are chosen as ϕ(0) = 0, ϕ(T) = π/2, θ(T) = π/2, (9) where T is final time moment, the L system will completely transfer to the level |3⟩ if initially in the level |2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Second, let us consider the R system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' We set its invariant as IR = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed 0 sin ϕ cos θ i cos ϕ sin ϕ cos θ 0 sin ϕ sin θ −i cos ϕ sin ϕ sin θ 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' (10) Similarly, we can obtain the eigenstates of this invariant IR: ��φR 0 � = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed sin ϕsinθ i cos ϕ − sin ϕ cos θ \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , ��φR± � = 1 √ 2 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed − cos ϕ sin θ ± i cos θ i sin ϕ cos ϕ cos θ ± i sin θ \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , (11) with corresponding eigenvalues µ0 = 0 and µ± = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' For the R system, we find that the parameter constraints and LR phases of R system are exactly the same as those of L system, as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' (5) and (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' This means that if we drive the L or R sys- tem to evolve along the eigenstate ��φL 0 � or ��φR 0 � , we can apply the same pulse scheme by inversely solving the constraint con- ditions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Now, we pay attention to the eigenstate ��φR 0 (t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' If we have the same boundary condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' (9), the R system will com- pletely transfer from |2⟩ to |1⟩ for t ∈ [0, T], which is completely different from the target energy level of the L system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' That is to say, we can apply the same pulse to a pair of L and R systems when they are initially at the level |2⟩ by choosing the invariant parameters to fulfill the boundary condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' This can drive the L system to fully evolve to the level |3⟩, while drive the R system to fully evolve to the level |1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Finally, their hand- edness can be determined by measuring their energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' As a result, the 100% chiral discrimination is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Letter Optics Letters 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='2 error amplitude α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='95 1 Fidelity Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' The systematic error amplitude α vs fidelity of different schemes: SPS (blue, dotted line), OSE (red, dashed line), and CD (green, solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Here, we consider a simple parameter scheme (SPS) to show how to achieve an efficient chiral discrimination by invariant- based inverse engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' When we choose ϕ(t) = πt 2T , θ(t) = π 2 , (12) to satisfy the boundary conditions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Inversely, we can get the parameters of Hamiltonian, from the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' (5), as Ω = π 2T , Ωq = π 2 cot πt 2T , (13) where T is pulse duration and t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 2, we plot the evolution curve of the level population of the L and R sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' It can be seen that the two systems are initially at the same level |2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' At t = T, the population of the L system com- pletely transfers to level |3⟩, while the population of the R sys- tem completely transfers to level |1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Therefore, through mea- suring their energy spectrum or population of the system, we can determine its chirality: if the population of the state |3⟩ is 1, this is a left-handed system, and if the population of the state |1⟩ is 1, it is a right-handed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' On the other hand, when we consider the influence of con- trol errors that may occur in the experiment on the fidelity (or discrimination) of the resolution scheme, it is necessary to op- timize the LRI scheme with respect to these errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' We use a new Hamiltonian H′ to indicate the existence of errors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=', H → H′ = H + He, where He is error Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' The fidelity is generally defined as F = ��� ψ(T) �� ψ′(T) ���2, (14) where |ψ(T)⟩ is target state and |ψ′(T)⟩ is actual state of system at the final moment T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Using perturbation theory [30], we have FL,R ≈ 1 − 1 ¯h2 ∑ ± ���� � T 0 dt � φL,R 0 (t) ��� He ���φL,R j (t) � eiηj(t) ���� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' (15) We first consider the influence of systematic error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' In this case, the error Hamiltonian is described as HL,R e = αHL,R, (16) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='5 1 error amplitude δ (1/T) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='95 1 Fidelity Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' The detuning error amplitude δ vs fidelity of different schemes: SPS (blue, dotted line), OSD (red, dashed line), and CD (green, solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' where, α is a dimensionless parameter, representing the ampli- tude of systematic error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' (15) and (16), we can get FL = FR = 1 − α2 ���� � T 0 ( ˙θ sin ϕ + i ˙ϕ)eiη+(t)dt ���� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' (17) Obviously, the fidelity of the target level for the L and R sys- tems is affected by the systematic error in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' There- fore, we only need to analyze the influence of error on the fi- delity of the L or R system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' The systematic error sensitivity is defined as qα = − ∂2FL,R 2∂α2 |α=0 = − ∂FL,R ∂(α2) |α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' (18) The smaller the sensitivity, the smaller the impact of error on fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Then we have qα = ���� � T 0 ( ˙θ sin ϕ + i ˙ϕ)eiη+(t)dt ���� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' (19) To meet the boundary conditions, we still choose ϕ(t) = πt 2T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' (20) We do not set the form of θ(t) at first, but try the Fourier series type of Ansatz with regard to the LR phase η+ η+(t) = −[n sin(3ϕ) − ϕ], (21) where n is a real number that can be chosen freely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' From the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' (8), the parameter θ(t) takes the form θ(t) = arccot3n cos(3ϕ) sin ϕ − sin ϕ + 1 3n cos(3ϕ) sin ϕ − sin ϕ − 1, (22) which satisfies the boundary condition θ(T) = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Based on the above equations, we can calculate the systematic error sensitivity qα numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' When n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='07, the systematic er- ror sensitivity reaches the minimum value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='52, which is defined as the optimal scheme for systematic error sensitiv- ity (OSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 3, we compare the influence of systematic error on the fidelity or discrimination with several coherent Letter Optics Letters 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Fidelity FL,R vs the systematic error amplitude α and detuning error amplitude δ by LRI scheme of n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' The yellow area in the middle corresponds to FL,R ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' control schemes, including OSS, SPS, and the counter-dabatic (CD) shortcut method in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' We can observe that all these schemes can achieve 100% discrimination in the absence of the error, and the OSE scheme is the most robust against systematic error, followed by SPS, and finally CD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Another important error in experiment is the detuning error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' In this case, the error Hamiltonian is HL,R e = δ¯h(|3⟩ ⟨3| − |1⟩ ⟨1|), (23) where δ represents the detuning amplitude, and its unit is 1/T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' In the same way, we can obtain the fidelity as FL = FR = 1 − δ2 4 ��� � T 0 [cos(2θ) sin(2ϕ) + 2i sin(2θ) sin ϕ]eiη+(t)dt ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' (24) And we have qδ = ���� � T 0 [cos(2θ) sin(2ϕ) + 2i sin(2θ) sin ϕ]eiη+(t)dt ���� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' (25) Here, the parameters ϕ and η+ are chosen as the same forms of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' (20) and (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' We can find that, the detuning error sen- sitivity reaches the minimum value 0 when n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' We call the corresponding parameter scheme as the optimal scheme for detuning error (OSD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 4, we compare the influence of detuning error on the fidelity or discrimination with OSD, SPS, and CD control schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Again, the OSD scheme is the most robust against the detuning error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Furthermore, we plot how the fidelity is affected by the systematic error and detuning er- ror in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' It can be seen that the optimal scheme shows high robustness against these two errors with a broad range of high efficiencies over 99% .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' In conclusion, we propose a highly efficient and robust chiral discrimination method for the cyclic three-level systems of chi- ral molecules based on the invariant-based inverse engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Through applying to the same pulse on the three-level system, molecules with different chirality will transit to different energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' The L system stay in |3⟩ and R system stay in |1⟩ at the final time from the same initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' We can realize the 100% chiral discrimination of molecules by measuring population or energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Moreover, we can design the corresponding optimization schemes with respect to different experimental er- rors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' By comparison, the optimization schemes are superior to the SPS and the CD schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' This study was supported by the National Natural Sci- ence Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 12004006, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 12075001, and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 12175001), Anhui Provincial Key Research and Development Plan (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 2022b13020004), and the Anhui Provincial Natural Science Foundation (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 2008085QA43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Disclosures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' The authors declare no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Data Availability Statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Data underlying the results pre- sented in this Letter are not publicly available at this time but may be obtained from the authors upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' REFERENCES 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Pasteur, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Chim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 24, 442 (1848).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Dashen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' D 3, 1879 (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Fu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Liu, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Hu, Compos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 160, 111 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Gu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Zhan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Zhang, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Bu, Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 45, 3122 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Leitereg, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Guadagni, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Harris, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Mon, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Teranishi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Agric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Food Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 19, 785 (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Hutt and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Tan, Drugs 52, 1 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Gal, Chirality 12, 959 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Shapiro, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Frishman, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Brumer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 84, 1669 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Vitanov and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Drewsen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 122, 173202 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Ye, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Chen, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Li, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' A 100, 043403 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Torosov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Drewsen, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Vitanov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' A 101, 063401 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Torosov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Drewsen, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Vitanov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 2, 043235 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Song, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Xia, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Su, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Jiang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' A 100, 043413 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Wu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Su, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Xia, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Jiang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Song, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Express 28, 33475 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Han, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Su, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Xia, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Jiang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Song, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 13, 044021 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Kang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Shi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Song, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Xia, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 45, 4952 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Ye, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Chen, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Li, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 2, 033064 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Guo, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Gong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Ma, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Shu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' A 105, 013102 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Vitanov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Yatsenko, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Bergmann, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' A 68, 043401 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Demirplak and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rice, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' A 107, 9937 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Berry, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' A 42, 365303 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Chen, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Lizuain, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Ruschhaupt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Guéry-Odelin, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Muga, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 105, 123003 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Song, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Ai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Qiu, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Deng, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' A 93, 052324 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Torosov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Guérin, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Vitanov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 106, 233001 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Genov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Schraft, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Halfmann, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Vitanov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 113, 043001 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Genov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Schraft, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Vitanov, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Halfmann, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 118, 133202 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Lewis and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Riesenfeld, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 10, 1458 (1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Chen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Ruschhaupt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Schmidt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' del Campo, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Guéry-Odelin, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Muga, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 104, 063002 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Chen and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Muga, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' A 86, 033405 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Ruschhaupt, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Chen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Alonso, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Muga, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 14, 093040 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Qi and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Jing, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' A 105, 053710 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Kang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Song, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Xia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Miranowicz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Zheng, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Nori, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 4, 013233 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Unanyan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Yatsenko, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Bergmann, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Shore, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 139, 48 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' +2 ude0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='9error 0 b00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='85etunin0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='8 20 litude0 amp rrortematic-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='2 sysLetter Optics Letters 5 FULL REFERENCES 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Pasteur, “On the Relations Crystalline Form, Chemical Composition and Direction of Polarization Rotatorie,” Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Chim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 24, 442–459 (1848).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Dashen, “Some features of chiral symmetry breaking,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' D 3, 1879 (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Fu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Liu, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Hu, “A novel category of 3d chiral material with negative Poisson’s ratio,” Compos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 160, 111–118 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Gu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Zhan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Zhang, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Bu, “Chiral chemistry of metal–camphorate frameworks,” Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 45, 3122–3144 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Leitereg, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Guadagni, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Harris, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Mon, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Teranishi, “Chemical and sensory data supporting the difference between the odors of the enantiomeric carvones,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Agric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Food Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 19, 785– 787 (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Hutt and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Tan, “Drug chirality and its clinical significance,” Drugs 52, 1–12 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Gal, “The discovery of stereoselectivity at biological receptors: Ar- naldo Piutti and the taste of the asparagine enantiomers—History and analysis on the 125th anniversary,” Chirality 12, 959–976 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Shapiro, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Frishman, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Brumer, “Coherently controlled asym- metric synthesis with achiral light,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 84, 1669 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Vitanov and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Drewsen, “Highly efficient detection and separa- tion of chiral molecules through shortcuts to adiabaticity,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 122, 173202 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Ye, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Chen, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Li, “Effective two-level models for highly efficient inner-state enantioseparation based on cyclic three- level systems of chiral molecules,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' A 100, 043403 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Torosov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Drewsen, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Vitanov, “Efficient and robust chi- ral resolution by composite pulses,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' A 101, 063401 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Torosov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Drewsen, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Vitanov, “Chiral resolution by composite Raman pulses,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 2, 043235 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Song, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Xia, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Su, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Jiang, “Robust and highly efficient discrimination of chiral molecules through three- mode parallel paths,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' A 100, 043413 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Wu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Su, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Xia, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Jiang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Song, “Discrimination of enantiomers through quantum interference and quantum Zeno effect,” Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Express 28, 33475–33489 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Han, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Su, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Xia, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Jiang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Song, “Two-Path Interference for Enantiomer-Selective State Transfer of Chiral Molecules,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 13, 044021 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Kang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Shi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Song, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Xia, “Effective discrimination of chiral molecules in a cavity,” Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 45, 4952–4955 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Ye, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Chen, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Li, “Fast enantioconversion of chiral mixtures based on a four-level double-∆ model,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 2, 033064 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Guo, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Gong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Ma, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Shu, “Cyclic three-level-pulse-area theorem for enantioselective state transfer of chiral molecules,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' A 105, 013102 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Vitanov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Yatsenko, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Bergmann, “Population transfer by an amplitude-modulated pulse,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' A 68, 043401 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Demirplak and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rice, “Adiabatic population transfer with con- trol fields,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' A 107, 9937–9945 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Berry, “Transitionless quantum driving,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' A 42, 365303 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Chen, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Lizuain, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Ruschhaupt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Guéry-Odelin, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Muga, “Shortcut to Adiabatic Passage in Two- and Three-Level Atoms,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 105, 123003 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Song, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Ai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Qiu, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Deng, “Physically feasible three- level transitionless quantum driving with multiple Schrödinger dynam- ics,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' A 93, 052324 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Torosov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Guérin, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Vitanov, “High-Fidelity Adiabatic Passage by Composite Sequences of Chirped Pulses,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 106, 233001 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Genov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Schraft, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Halfmann, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Vitanov, “Correction of Arbitrary Field Errors in Population Inversion of Quantum Systems by Universal Composite Pulses,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 113, 043001 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Genov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Schraft, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Vitanov, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Halfmann, “Arbitrarily Accurate Pulse Sequences for Robust Dynamical Decoupling,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 118, 133202 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Lewis and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 10, 1458–1473 (1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Chen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Ruschhaupt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Schmidt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' del Campo, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Guéry-Odelin, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Muga, “Fast Optimal Frictionless Atom Cooling in Harmonic Traps: Shortcut to Adiabaticity,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 104, 063002 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Chen and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Muga, “Engineering of fast population transfer in three-level systems,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' A 86, 033405 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Ruschhaupt, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Chen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Alonso, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Muga, “Optimally robust shortcuts to population inversion in two-level quantum systems,” New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 14, 093040 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Qi and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Jing, “Accelerated adiabatic passage in cavity mag- nomechanics,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' A 105, 053710 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Kang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Song, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Xia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Miranowicz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content='- B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Zheng, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Nori, “Nonadiabatic geometric quantum computation with cat-state qubits via invariant-based reverse engineering,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 4, 013233 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Unanyan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Yatsenko, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Bergmann, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Shore, “Laser-induced adiabatic atomic reorientation with control of diabatic losses,” Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} +page_content=' 139, 48–54 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE2T4oBgHgl3EQfRQcf/content/2301.03778v1.pdf'} diff --git a/AtAzT4oBgHgl3EQf__8r/content/tmp_files/2301.01955v1.pdf.txt b/AtAzT4oBgHgl3EQf__8r/content/tmp_files/2301.01955v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e8762767e927374daae7b2e94b2a17c851f11d10 --- /dev/null +++ b/AtAzT4oBgHgl3EQf__8r/content/tmp_files/2301.01955v1.pdf.txt @@ -0,0 +1,1595 @@ +Adaptively Clustering Neighbor Elements for Image Captioning +Zihua Wang1,2 +Xu Yang1 +Haiyang Xu2* +Hanwang Zhang3 +Chenliang Li2 +Songfang Huang2 +Fei Huang2 +Yu Zhang1* +1 School of Computer Science & Engineering, Key Lab of Computer Network +& Information Integration (Ministry of Education), Southeast Univ., Nanjing, China +2Alibaba Group +3 School of Computer Science & Engineering, Nanyang Technological Univ., Singapore. +{zihua, 101013120, zhang yu}@seu.edu.cn,{shuofeng.xhy, lcl193798, +songfang.hsf, f.huang}@alibaba-inc.com, hanwangzhang@ntu.edu.sg +Abstract +We design a novel global-local Transformer named Ada- +ClustFormer (ACF) to generate captions. We use this name +since each layer of ACF can adaptively cluster input el- +ements to carry self-attention (Self-ATT) for learning lo- +cal context. Compared with other global-local Transform- +ers which carry Self-ATT in fixed-size windows, ACF can +capture varying graininess, e.g., an object may cover dif- +ferent numbers of grids or a phrase may contain diverse +numbers of words. To build ACF, we insert a probabilis- +tic matrix C into the Self-ATT layer. +For an input se- +quence {s1, ..., sN}, Ci,j softly determines whether the +sub-sequence {si, ..., sj} should be clustered for carrying +Self-ATT. For implementation, Ci,j is calculated from the +contexts of {si, ..., sj}, thus ACF can exploit the input itself +to decide which local contexts should be learned. By us- +ing ACF to build the vision encoder and language decoder, +the captioning model can automatically discover the hid- +den structures in both vision and language, which encour- +ages the model to learn a unified structural space for trans- +ferring more structural commonalities. The experiment re- +sults demonstrate the effectiveness of ACF that we achieve +CIDEr of 137.8, which outperforms most SOTA captioning +models and achieve comparable scores compared with some +BERT-based models. The code will be available in the sup- +plementary material. +1. Introduction +Image Captioning (IC) aims to learn a shared vision- +language representation space for facilitating the transfer of +multimodal knowledge to generate visually grounded sen- +*Corresponding authors. +tence [22]. Two prevailing deep learning techniques help +the IC model learn such space. +The first one is the vi- +sion encoder-language decoder pipeline [41] which back- +propagates the language semantic to the visual encoder +and another one is the attention mechanism [46] which di- +rectly bridges between vision and language domains for +transferring multimodal knowledge. +Transformers [39], +which build the encoder and decoder based on dense at- +tention operations, have both of the above-mentioned ad- +vantages. Transformers have two types of attention opera- +tions which are self-attention (Self-ATT) and cross-modal +attention (Cross-ATT). From the perspective of structure +learning, Self-ATT applies the fully connected (FC) graph +prior to the data sequence. +By using Self-ATT in both +encoder and decoder, the graph structures of both vision +and language data can be discovered and Cross-ATT helps +transfer these structural commonalities for narrowing the +modality gaps. +Therefore, Transformer prevails in IC +tasks [10,12,13,28]. +Interestingly, structure learning is one of the most sig- +nificant research directions of IC since the paired vision- +language data usually share a unified internal semantic +structure although they have diverse external appearances. +Thus, if this unified semantic structure is captured, more +structural commonalities can be transferred for generating +better captions. Motivated by this, various IC models are +proposed to exploit scene graphs [5, 21, 49] or hierarchy +trees [43, 51] to narrow the domain gap. However, such +structures need additional well-trained parsers. Moreover, +vision and language parsers usually have domain gaps that +the parsed structures of the paired image-sentence may not +match, which may even weaken the effectiveness of these +IC models. We prefer an IC model that can adaptively dis- +cover the unified semantic structures to remove the costs of +the additional structure annotations and more importantly, +arXiv:2301.01955v1 [cs.CV] 5 Jan 2023 + + (a) Fixed-Size Transformer +s1 s2 s3 s4 s5 s6 s7 +s8 +Input +1-st +layer +2-nd +layer +3-rd +layer +(b) ACF +s1 s2 s3 s4 s5 s6 s7 s8 +Input +1-st +layer +2-nd +layer +3-rd +layer +… +… +… +… +(c) ACF-based IC +riding +a snow board +on +snow +A man +riding a snow board +on snow +A man riding a snow board on snow. +riding +a +A +man +snow +board +on +snow +A man +Figure 1. (a) Transformer with fixed-size windows (size = 2); (b) +ACF which adjusts the window size according to the input. (c) +ACF-based IC. The left/right part shows how the vision/language +ACFs cluster image grids/language words for transferring struc- +tural commonalities. +to learn a unified structure space for transferring structural +commonalities. +Transformer seems to be a good starting point since +it can implicitly build graphs by Self-ATT. However, it +exploits the FC graph prior, while the useful semantic +structure is usually sparse and hierarchical like the scene +graphs or trees. +To discover more sparse structures, re- +searchers design various global-local Transformers [20,29, +33]. As sketched in Figure 1(a), these Transformers grad- +ually merge the neighbor elements in fixed-size windows +into bigger clusters and carry Self-ATT in each cluster. For +example, the 1-st layer clusters 2 neighboring elements like +{s1, s2} to carry Self-ATT for local contexts and the 2- +nd layer merges {s1, s2} and {s3, s4} into a bigger one +to learn more global context. +Then a hierarchical struc- +ture is built from lower to higher layers where local and +global contexts are respectively captured. However, these +Transformers still do not satisfy our requirement since vi- +sion and language data have diverse graininess, e.g., objects +may cover varying grids and phrases may compose different +numbers of words, while fixed-size windows cannot effec- +tively capture such varying graininess. +To capture the varying graininess, we propose to +Adaptively +Cluster +the +neighbor +elements +to +carry +Self-ATT and named the novel Transformer as Ada- +ClustFormer (ACF). As shown in Figure 1(b), in each +layer, the window size is not fixed but can be adjusted +to each specific input sequence, e.g., in the 1-st layer, +{s1, s2, s3}, {s4}, {s5, s6}, {s7}, {s8} are respectively +clustered. The higher layers merge small clusters into big- +ger ones for global contexts, e.g., the 2-nd layer respectively +merges {s1, s2, s3, s4, s5, s6}, {s7, s8} into two clusters to +carry Self-ATT. To achieve this adaptive clustering, we in- +sert a probabilistic clustering matrix C into the Self-ATT +layer, where the probability Cij softly determines whether +the sub-sequence {si, ..., sj} should be clustered or not. To +calculate Cij, we consider whether the next element sj is +similar to the mean-pooling of {si, ..., sj−1}. Thus ACF +can adjust the window of Self-ATT based on each specific +data sample. +To construct an IC model based on ACF, besides build- +ing 1-D ACF for the language decoder, we also extend it +to the 2-D ACF as the vision encoder. In this way, both +the visual encoder and language decoder can automatically +discover the hidden structures of the image and language +data. This means that the ACF model does not need any +additional structure annotations as some previous IC mod- +els [2, 5] but still exploits the sparse structures implied in +both vision and language data. For example, as shown in +Figure 1(c), a visual ACF can merge the smaller grids into +bigger regions to capture both grid-level [15] and region- +level [4] contexts. And the language one gradually clus- +ters the single words into phrases to generate the captions +in an imaginary phrase-by-phrase manner [38, 48]. More +importantly, compared with certain global-local Transform- +ers which are exclusively developed in vision and language +domains [24, 47], the visual and language ACF exploit the +same way to discover hidden structures. So, our ACF model +is a homogeneous structure that helps transfer more struc- +tural commonalities between vision and language domains, +e.g., as shown in Figure 1(c), the patches of the object “snow +board” is clustered in the image and correspondingly, the +phrase “a snow board” is also clustered in the language do- +main. +In summary, our contributions can be listed as follows: +• We propose ACF that can adaptively capture varying +graininess. +• We extend ACF to the 2-D case for building a homo- +geneous IC model that learns unified structural space +for transferring more structural commonalities. +• The experimental results show that our ACF model +outperforms the classic Transformers in IC. +2. Related Work +Image Captioning (IC). IC aims to generate descriptions +according to the given images. +Typically, an encoder- +decoder paradigm is used to convert visual inputs to se- +quence outputs. In the early stage, image features are ex- +tracted by CNN-based encoders, as the input of the RNN- +based decoders [4, 16, 35, 41]. For example, Up-Down [4] +employs a Faster R-CNN [34] to extract image region fea- +tures and LSTM networks to generate sentences. +Nowadays, Transformer-based models have shown their + +might in Neural Language Process (NLP) and replace RNN- +based decoders in IC [12, 14, 19]. Subsequently, more ad- +vanced Transformer-based decoders are proposed, e.g., M2 +Transformer [8] proposes a meshed-memory Transformer +to interact with the low-level and high-level features; X- +Linear Transformer [31] selectively capitalizes the visual +information from image regions by bilinear pooling. +However, these models still use CNN-based feature ex- +tractors. +More recently, witnessing the boom of Vision +Transformers (ViT) [9, 24], researchers use ViT-based vi- +sual encoders for captioning. For instance, CPTR [23] in- +troduces grid-based features that are extracted by ViT [9] +instead of using the ROI-based features; DLCT [25] fuses +the ROI-based features with the grid-based features to over- +come the shortcoming of both features. +Besides that, +some models exploit the knowledge distilled from Vision- +Language BERTs for better captions [18]. +VinVL [52] +and GRIT [28] propose the object detection model in IC. +ClipCAP [27] and LEMON [13] introduce large-scale pre- +training on IC. Noteworthy, the methods above employ the +ViT [9] or Swin Transformer [24] as their backbone. Thus, +our ACF adopts the Swin Transformer as our encoder back- +bone. +Among the previous IC models, Auto-Parsing Network +(APN) [48] has a similar motivation as ours, which also in- +serts a clustering matrix into the Self-ATT layer. However, +Ada-ClustFormer (ACF) calculates this matrix differently. +APN only considers whether pairwise neighboring elements +should be clustered or not, while we calculate this proba- +bility from a more global scope. Specifically, we consider +whether the next element is similar to the previous clustered +elements. More importantly, we extend our ACF into the 2- +D case, which can adaptively cluster the visual patches into +regions, while APN only treats a sequence of ROI features +as the visual input and still applies a 1-D clustering matrix +to address it. More comparisons will be given in the supple- +mentary material. +Global-Local Transformer. +To alleviate the fully con- +nected graph prior in Transformer, researchers propose var- +ious global-local Transformers to learn sparse structures of +the language [6, 26]. For example, Global-local [26] intro- +duces a fixed-size of the global and local attention model in +neural machine translation. Longformer [6] proposes global +and local window attentions, which can provide inductive +bias and long sequence representation, respectively. +Hi- +Transformer [44] learns sentence-level and document-level +semantics through the hierarchical structure. +The global-local Transformer mechanism is also effec- +tive in vision area [7, 25, 53]. Pairwise and patchwise self- +attention are proposed in image recognition [53]. Further- +more, GLiT [7] proposes to adaptively trade off the global +and local information of the images. DLCT [25] explores +the global and local information by the combination of grid- +based features and ROI-based features. +However, these models are exclusively developed in a +single domain (either NLP or CV), while our ACF provides +a general approach in both the vision and language domains. +Thus, using ACF to build the IC model encourages learn- +ing a unified structure space for transferring more structure +commonalities. +3. Ada-ClustFormer IC model +Compared +with +the +classic +Transformer, +Ada- +ClustFormer +(ACF) +inserts +an +adaptively +clustering +matrix C into each self-attention (Self-ATT) layer to +adaptively control the scope of Self-ATT. The calculation +of C is detailed in Section 3.1 where we first show the 1-D +language case and then extend it to the 2-D vision case. By +stacking these revised Self-ATT layers, ACF can be built +for constructing the vision encoder and language decoder +for captioning (cf. Section 3.2). +3.1. Ada-ClustFormer +Multi-Head Attention (MHA). ACF is built based on +Transformer, whose most elemental building block is the +Multi-Head Attention (MHA). Given the query Q +∈ +RNQ×d, key K ∈ RNK×d, and value V ∈ RNV ×d, MHA +calculates the output Z = MHA(Q, K, V) as: +Input: +Q, K, V +ATT: +Al = Softmax(QWQ +l (KWK +l )T +√ +d +) +Head : +Hl = AlVWV +l , +Multi-Head: +H = [H1, H2, ..., Hh]WH, +Output: +Z = LN(H + Q), +(1) +where WQ +l , WK +l , WV +l +∈ Rd×dh, WH +l +∈ Rd×d are all learn- +able parameters; h denotes the head number and dh = d/h; +Al is the l-th attention matrix corresponding to the l-th head +Hl; [·] is the concatenation operation; and LN denotes to the +Layer Normalization. +Given an input sequence S = {s1, ..., sN}, if Q = +K = V = S, Eq. (1) is also called self-attention (Self- +ATT). Self-ATT captures the global contexts between any +two elements si and sj by calculating the pairwise atten- +tion weight in the “ATT” operation. From the perspective +of structure learning [5], single-head Self-ATT constructs +a fully-connected (FC) graph where the nodes are the ele- +ments of S and the pairwise edges are weighted by the pair- +wise attention weight. Correspondingly, a h-head Self-ATT +constructs h FC graphs with different edge weights. +Adaptive Clustering Matrix C. To sparsify this FC-graph, +researchers [9, 24] propose to carry Self-ATT in fixed-size +windows, which is achieved by revising “Head” in Eq. (1): +C-based Head : +H = Softmax(A ⊗ C)VWV , +(2) + +where “⊗” denotes the element-wise production; C is a +N × N binary clustering matrix that only the elements +in the window can attend to each other, i.e., if the win- +dow size is w, Ci,j = 1 if |i − j| ≤ w and Ci,j = 0 +if |i − j| > w. However, language or vision data usually +have diverse graininess, e.g., a phrase may contain different +numbers of words or an object may cover diverse spatial +regions, while the fixed-size windows can not capture the +varying graininess. +To amend this, we revise the binary C to a probabilistic +one where Ci,j softly determines whether to cluster the em- +beddings from si to sj for carrying Self-ATT. Then if Ci,j +is small, the pairwise attention in A between si and sj is +weakened in Eq. (2), which means si and sj are less likely +to stay in the same cluster. To adaptively decide the win- +dow size according to each specific input for capturing the +varying graininess, we use the input itself to calculate Ci,j: +Ci,j = P(si, ..., sj) = +j� +k=i +P(sk|si, ..., sk−1), +(3) +where the joint distribution is decomposed to the produc- +tions of conditional distributions P(sk|si, ..., sk−1), which +softly decides whether to merge a new element sk into +the sub-sequence {si, ..., sk−1}. +In the implementation, +P(sk|si, ..., sk−1) is calculated as: +P(sk|si, ..., sk−1) = Sigmoid(FC([sk, si:k−1])), +(4) +where si:k−1 is the mean pooling of the embeddings from +si to sk−1. Intuitively, Eq. (4) exploits the context of the +whole sub-sequence {si, ..., sk−1} to decide whether to +merge a new element {sk} into this sub-sequence. Note +that Eq. (3) and Eq. (4) only make sense when i < k. Since +clustering the embeddings from si to sk equals to cluster- +ing from sk to si, we set Ci,k = Ck,i if i > k and since a +single element si is itself a cluster, we set Ci,i = 1. +From Eq. (3), we can also find that: +Ci,j =P(sj|si, ..., sj−1) × P(si, ..., sj−1) +=P(sj|si, ..., sj−1) × Ci,j−1. +(5) +Since P(sj|si, ..., sj−1) ≤ 1, we have Ci,j ≤ Ci,j−1, +which means that two elements in the shorter distance are +more likely to be clustered for carrying Self-ATT. In this +way, local contexts are encouraged to be captured, as is +shown in Figure 2(a). +Stacking Revised Self-ATT. To learn global contexts, we +can stack these revised Self-ATT layers. When stacking, +we hope that the higher layers will carry Self-ATT in bigger +windows than the lower layers to capture the global con- +texts [43, 48]. To achieve this, for the m-th layer, we re- +calculate C(m) as ˜C(m): +˜C(m) = (1 − C(m)) ˜C(m−1) + C(m). +(6) +s1 s2 s3 s4 s5 s6 +s1 s2 s3 s4 s5 s6 +C1,4 = C1,3 × P( s4 | s1, s2, s3) +Sigmoid(FC([s4, s1:s3])) +(a) Calculation of C1,4 +(b) C(2) ≥ C(1) +1-st +layer +2-nd +layer +~ +~ +Figure 2. (a) shows how to calculate C1,4, where the shade denotes +the probability value, the darker the color, the larger the probability +value. (b) shows that the clustered elements in the lower layer will +be further clustered in a higher layer, e.g., the color of {s1, s2, s3} +in the 2-nd layer is darker than the 1-st layer. +Horizontal +Upsampling +(a) Calculation of C1,4;1,3 +(b) Down-up Sampling Strategy +Ph(v1;1, ..., v4;1) +Pv(v1;1, ..., v1;3) +s1 s2 +s4 +s3 +s1 s2 +C1,2 +C2,3 +∏ +∏ +C1,4;1,3 +Horizontal +Upsampling +(a) Calculation of C1,4;1,3 +(b) Down-up Sampling Strategy +Ph(v1;1, ..., v4;1) +Pv(v1;1, ..., v1;3) +s1 s2 +s4 +s3 +s1 s2 +C1,2 +C2,3 +∏ +∏ +C1,4;1,3 +Figure 3. (a) The example of 2-D C, where C1,4;1,3 is used as +the example, which is decomposed into vertical and horizontal di- +rections probabilities. (b) Overview of the Down-Up Sampling +Strategy. +Then ˜C(m) is used in Eq. (2) when m > 1 and ˜C(1) = +C(1). Since 0 ≤ C(m) +i,j +≤ 1, ˜C(m) +i,j +is a convex combination +of ˜C(m−1) +i,j +and 1, which means that ˜C(m−1) +i,j +≤ ˜C(m) +i,j +≤ 1. +If ˜C(m−1) +i,j +is large, i.e., the sub-sequence {si, ..., sj} should +be clustered in the (m − 1)-th layer, then ˜C(m) +i,j +must be +larger, i.e., {si, ..., sj} is also clustered in the m-th layer. +For example, Figure 2(b) shows that the 2-nd layer will +further cluster {s1, s2, s3} since ˜C(1) +1,3 ≤ ˜C(2) +1,3. Thus, the +higher layers will carry Self-ATT in a bigger window than +the lower layers to learn more global contexts. +2-D Clustering Matrix. Eq. (3) shows how to calculate +C when the input is a 1-D language sequence, next we +extend it to the 2-D vision surface. +Given a 2-D fea- +ture map V += {v1,1, ..., vH,W }, we use Ci,j;x,y to de- +note the probability that softly decides whether a sub-region +{vi,x, ..., vj,y} should be clustered or not, which is: +Ci,j;x,y = P(vi;x, ..., vj;y) += +j +� +k=i +y +� +u=x +P(vk;u|vi;x, vi+1;x, ..., vk−1;u−1) +(7) +where i, j and x, y respectively denote the horizontal and +vertical dimensions. To cover all the sub-regions in a H×W + +Image +Self-ATT +Add&LN +1-D C +Self-ATT +Add&LN +Words +Cross-ATT +Add&LN +Captioning: Z +me× +Encoder +Decoder +md× +Q,K,V +Q,K,V +K,V +Q +2-D C +Figure 4. Overview of our ACF-based encoder-decoder IC model. +The “Add&LN” is the Add and Layer Normalization. me/md rep- +resent the number of the encoder/decoder layers, respectively. +map, it requires applying O(H2 × W 2) times for Eq. (4) to +get all the probabilities. To reduce the computation burden, +we apply the independence assumption to decompose the +2-D distribution into two independent ones, which respec- +tively correspond to the horizontal and vertical dimensions: +P(vi;x, ..., vj;y) = Ph(vi;x, ...vj;x)Pv(vi;x, ..., vi;y) += +j +� +k=i +Ph(vk;x|vi;x, ..., vk−1;x) +y +� +u=x +Pv(vi;x|vi;x, ..., vi;u−1), +(8) +In this way, we only need to apply O(H2 + W 2) times +for Eq. (4) and once matrix production. +Noteworthy, as +sketched in Figure 2, for the 2-D region which spans the +horizontal axis from i to j and the vertical axis from +x to y, we use the left-most vertical and top-most hor- +izontal to calculate two 1-D distributions and then mul- +tiply them to get Ci,j;x,y. +As Figure 3(a) shows, to +calculate C1,4;1,3, for the vertical distribution Pv, the +horizontal ordinate is fixed to 1 and the vertical or- +dinate changes. +Ph(vk;1|v1;1, ..., vk−1;1)|k=1,2,3,4 and +Pv(v1;u|v1;1, ..., v1;u−1)|u=1,2,3 are calculated in the same +way as Eq. (4). The above-mentioned symmetric character- +istic is also applied. +Down-Up Sampling Strategy. +If the sequence (feature +map) is too long (big), we can apply the Down-Up Sam- +pling Strategy to reduce the computation cost. We use a 1-D +language case as an example to show this strategy. For S = +{s1, ..., sL}, we can downsample it to ¯S = {¯s1, ..., ¯sL/2} +where ¯si is the mean pooling of s2∗i−1 and s2∗i. Then ¯S +is used in Eq. (3) and Eq. (4) to get ¯ +C. To upsample ¯C to +the original size, we set Ci,j = ¯ +C⌈i/2⌉,⌈j/2⌉. Figure 3(b) +shows one simple case where L = 4. +3.2. Encoder-Decoder Architecture +As is shown in Figure 4, we apply the ACF to build the +vision encoder and language decoder. Compared to the clas- +sic Transformer, our ACF introduces clustering-restrained +attention head. Specifically, in encoder, we calculate a 2-D +clustering matrix C (cf. Eq. (7)) to softly cluster the ele- +ments for carrying Self-ATT. Similarly, in decoder, the at- +tention head is revised with the 1-D C (cf. Eq. (5)). The +output of this encoder-decoder is used to calculate the word +distributions Z. +To train our IC model, we optimize the model by min- +imizing the cross-entropy loss and maximizing the Rein- +forcement learning (RL) [35] reward. First, we train the +model by minimizing the cross-entropy loss: +LCE = − log P(Z∗), +(9) +where Z∗ is the ground-truth captions. Then, we further +train the model by minimizing the negative reward: +Lrl = −EZs∼P (Z)(S(Z∗, Zs)), +(10) +where Zs is sampled from Z, E represents the mathemat- +ical expectation, and S represents the evaluation metrics, +e.g., CIDEr [40]. +4. Experiments +4.1. Dataset, Metrics, and Settings +MSCOCO. Following [8, 12, 14, 31, 48], we train and +evaluate our model on MSCOCO [22], which contains +123, 287 images, and each one is annotated with 5 cap- +tions. +In the experiments, we use the Karpathy split +(113,287/5,000/5,000 train/val/test images) [16] for offline +training and the official split (40775 test images) for online +testing. +Metrics. +We adopt five widely-used metrics in caption- +ing for evaluation, including BLEU [32], METOR [1], +ROUGE-L [36], CIDEr [40], and SPICE [3]. +Settings. In the training process, we convert all the captions +into lowercase and delete all the words that occur less than +6 times. The remaining 9487 words are regarded as our +vocabulary. We adopt Swin Transformer [24] as the visual +encoder to extract the visual features. The size of the feature +map is H × W = 12 × 12, and we apply the Down-Up +Sampling Strategy (cf. Section 3.1). We train 20/25 epochs +in the cross-entropy/RL stage. In the cross-entropy stage, +the Adam optimizer is used with the learning rate of 5 × +10−5 and decays by 0.8 per 5 epochs. In the RL stage, the +learning rate is initialized to 5 × 10−6 and we implement +the same decay policy for 10 epochs. Then the “Reduce- +On-Plateau” strategy is applied with a decay rate of 0.5 and +patience of 3. The batch size is 40 at the whole training +stage. + +Table 1. Comparison between with and without Ada-ClustFormer. +Models +me +md +B@4 +M +R +C +S +BASE +6S +6S +40.0 +29.7 +59.6 +134.4 +23.4 +ACF 1 +6C +6S +40.3 +29.6 +59.6 +134.7 +23.5 +ACF 2 +6S +6C +40.2 +29.8 +59.9 +135.1 +23.7 +ACF +6C +6C +41.1 +30.1 +60.2 +137.8 +24.1 +4.2. Ablation Studies +We conduct extensive ablations for quantifying the dif- +ference between classic self-attention (Self-ATT) layers and +Ada-ClustFormer (ACF) layers (cf. Section 4.2.1), the im- +pact of the depth of the ACF layers (cf. Section 4.2.2), and +the impact of the orders of ACF and the Self-ATT layers (cf. +Section 4.2.3). +4.2.1 +Differences Between ACF and Self-ATT +Comparing Methods. +To evaluate the effectiveness of +the ACF, we ablate our ACF with the following baselines: +BASE: We employ 6 Self-ATT encoder layers and de- +coder layers, which is shown in Table 1 as “6S”. ACF 1 +/ ACF 2: We replace the encoder/decoder with our ACF, +which is represented as “6C”. +Results. The results of the ablation are listed in Table 1. +Compared with BASE, we can find that only using ACF +encoder (ACF 1) or decoder (ACF 2) has marginal im- +provements, which is 0.3 or 0.7 on CIDEr. However, when +combining the ACF encoder and decoder to build a homo- +geneous architecture ACF, the improvement is substantial, +which is 3.4. This comparison suggests that a homogeneous +model transfers more structural commonalities for better +captions. +4.2.2 +Impact of the Layer Depth +Comparing Methods. ACF 3: We reduce the depth of the +encoder and decoder layer to 3. ACF 4/ACF 5: The num- +ber of the encoder/decoder layers is set to 3 and the number +of the decoder/encoder layer remains 6. +Results. From Table 2, we observe that stacking 6 layers +generally outperforms the 3-layer case. Our method with +6 ACF layers in the encoder and decoder achieves the best +performance among them. We also further explore the in- +fluence of me by fixing md = 6. We present the impact of +the number of the encoder layers me in Figure 5. It sug- +gests that CIDEr approximately linearly increases when me +increases. +4.2.3 +Impact of the Layer Order +Comparing Methods. We discuss the combination of the +ACF layers and the Self-ATT layers. We freeze the depth +Table 2. The performances with different layer depth +Models +me +md +B@4 +M +R +C +S +ACF 3 +3C +3C +38.9 +28.4 +58.8 +132.3 +22.0 +ACF 4 +6C +3C +39.3 +28.9 +59.1 +135.9 +23.7 +ACF 5 +3C +6C +40.2 +29.8 +59.7 +136.0 +24.0 +ACF +6C +6C +41.1 +30.1 +60.2 +137.8 +24.1 +Table 3. The impact of the layer orders. +Models +me +md +B@4 +M +R +C +S +ACF 5 +3C +6C +40.2 +29.8 +59.7 +136.0 +24.0 +ACF 6 +3C+ 3S +6C +40.7 +29.7 +59.9 +135.7 +23.8 +ACF 7 +3S+ 3C +6C +40.5 +29.9 +59.9 +136.1 +23.9 +ACF 2 +6S +6C +40.2 +29.8 +59.9 +135.1 +23.7 +ACF +6C +6C +41.1 +30.1 +60.2 +137.8 +24.1 +of the decoder layer md = 6 and quantify the influence of +the order of the encoders: ACF 5: It stacks 3 ACF lay- +ers. ACF 6/ACF 7: Both of them have 3 ACF layers and +3 Self-ATT layers. +The difference between them is that +ACF 7 encodes on 3 Self-ATT layers firstly. +Results. The results are listed in Table 3, where we can see +that the performances are not sensitive to the orders of ACF +and Self-ATT layers, i.e., ACF 6 and ACF 7 differ only 0.4. +We can also find that replacing all the Self-ATT layers with +our ACF layers will achieve the best captioning quality. +3 +4 +5 +6 +Number of encoder layers +136.0 +136.5 +137.0 +137.5 +138.0 +CIDEr +135.97 +136.6 +137.5 +137.83 +Figure 5. Impact of the number of encoder layers me. +Qualitative Results. We visualize the hierarchical struc- +tures of the image and the generated captions in Figure 6 +according to the 2-D and 1-D clustering matrix calculated +from the 1-st, 3-rd, 5-th, and 6-th layers in encoder and de- +coder. By inspecting the images and captions, we can find +that the patches and the words are respectively clustered, +e.g., in the left part of (b), the patches in the “motorcycles” +region are clustered, and in the right part, the words “sit- +ting on motorcycles” are clustered into a phrase. More im- +portantly, when uniting the image and caption, we can find +that structural commonalities are transferred, e.g., in (b), +the “motorcycle” region helps generate the phrase “sitting +on motorcycles”. + +A woman standing on the door of a train with a suitcase. +a woman +standing on +the door of a train +with a suitcase +standing on +a woman +the door of +a +a +train with +suitcase +Ground-truth: A woman in white and +black dress with suitcase on train. +BASE: A woman standing with a +suitcase. +ACF: A woman standing on the door of +a train with a suitcase. +Two people sitting on motorcycles next to a stop sign. +sitting on +Two people +motorcycles +a +next to +stop +sign +sitting on motorcycles +next to a stop sign +Ground-truth: Two people riding +motorcycles on a city street. +BASE: Two people riding black +motorcycles. +ACF: Two people sitting on +motorcycles next to a stop sign. +Ground-truth: A man with a hat and +eye glasses holding a cell phone. +BASE: A man with a cowboy hat +holding a cell phone. +ACF: A man wearing a cowboy hat +taking a picture with a cell phone. +A man wearing a cowboy hat taking a picture with a cell phone. +wearing +cowboy hat +a +taking +cell +picture with +wearing a cowboy hat +a man +taking a picture +a man +a +a +phone +with a cell phone +Two +people +(b) +(c) +(a) +taking a picture with a cell phone +A man wearing a cowboy hat +Two people sitting on motorcycles +next to a stop sign +a woman +standing on the door of a train +with a suitcase +Figure 6. Examples of the generated captions by BASE and ACF models. We visualize the 2-D C and 1-D C in the 1-st, 3-rd, 5-th, and +6-th layers as the clustered patches. +4.3. Comparisons with SOTA +Comparing Methods. Nowadays, the SOTA of image cap- +tioning has been updated quickly and these models can +be categorized into 3 groups. The first one is the meth- +ods which use ROI-based features, including Up-Down [4], +ORT [12], AoANet [14], M2 Transformer [8], Tree- +Transformer [43], APN [48], and X-Transformer [31]. +Among the above methods, Up-Down [4] deploys a famous +architecture with a CNN-based encoder and an LSTM- +based decoder. +ORT [12] applies Transformer to lan- +guage decoder. +AoANet [14] and M2 Transformer [8] +further improve the attention mechanism on the language +decoder. Tree-Transformer [43] and APN [48] reveal the +validity of the utilization of the sequence structure. +To +capture high-order interaction between sequence and re- +gions, X-Transformer [31] introduces a bilinear pooling +structure. The second group are the methods using grid- +based features: CPTR [23], Dual-Global [45], DLCT [25], +and PureT [42]. +Among them, Dual-Global [45] and +DLCT [25] combine the grid-based features with the ROI- +based features. +PureT [42] end-to-end trains the whole +model and PureT-standard/PureT-Swin respectively use +Transformer [9]/Swin Transformer [24] as the vision en- +coder to deal with the visual features, which is also ex- +tracted from a Swin Transformer. +The third group dis- +tills the knowledge from large-scale pretraining models: +RSTNet [54], and ViTCAP [10]. Accordingly, we seg- +ment the performances into 3 parts in Table 4, where the +top/middle/bottom parts are the ROI-based, grid-based, and +BERT-based models. Note that for APN, besides reporting +the results in their paper [48], which is got by using ROI- +based features, we also report the performances using the +same visual features as ours, which is denoted as “APN♯”. +Results. +From Table 4, we can see that ACF is com- +parable to most of state-of-the-art performance when +compared with ROI and grid-based models. +Moreover, + +STOPS +OPSTOPSTOPTable 4. The performances of SOTA methods on MSCOCO Karpathy split. +Models +Cross-Entroy Loss +CIDEr optimization +B@4 +M +R +C +S +B@4 +M +R +C +S +ROI-based feature +Up-Down [4] +36.2 +27.0 +56.4 +113.5 +20.3 +36.3 +27.7 +56.9 +120.1 +21.4 +ORT [12] +35.5 +28.0 +56.6 +115.4 +21.2 +38.6 +28.7 +58.4 +128.3 +22.6 +AoANet [14] +37.2 +28.4 +57.5 +119.8 +21.4 +38.9 +29.2 +58.8 +129.8 +22.4 +M2 Transformer [8] +- +- +- +- +- +39.1 +29.2 +58.6 +131.2 +22.6 +CATT [50] +37.3 +28.5 +57.4 +119.0 +21.5 +39.4 +29.3 +58.9 +131.7 +22.8 +APN [48] +- +- +- +- +- +39.6 +29.2 +59.1 +131.8 +23.0 +X-Transformer [31] +38.2 +28.8 +58.0 +122.0 +21.9 +39.7 +29.5 +59.2 +132.8 +23.2 +Grid-based feature +CPTR [23] +- +- +- +- +- +40.0 +29.1 +59.4 +129.4 +− +APN♯ [48] +- +- +- +- +- +40.1 +29.4 +59.4 +133.2 +23.3 +Dual-Global [45] +- +- +- +- +- +40.3 +29.2 +59.4 +132.4 +23.3 +DLCT [25] +- +- +- +- +- +40.8 +29.9 +59.8 +137.5 +23.3 +End-to-End training +PureT-standard [42] +- +- +- +- +- +40.3 +29.9 +59.9 +137.5 +23.8 +PureT-Swin [42] +- +- +- +- +- +40.9 +30.2 +60.1 +138.2 +24.2 +Visual-language BERT pretraining +RSTNet [54] +- +- +- +- +- +40.1 +28.9 +59.5 +135.6 +23.3 +ViTCAP-small [10] +35.7 +28.8 +57.6 +121.8 +22.1 +40.1 +29.4 +59.4 +133.1 +23.0 +ViTCAP-large [10] +36.3 +29.3 +58.1 +125.2 +22.6 +41.2 +30.1 +60.1 +138.1 +24.1 +ACF +38.1 +28.8 +58.4 +123.8 +21.8 +41.1 +30.1 +60.2 +137.8 +24.1 +Table 5. The scores on the MSCOCO online test server. +Models +B@4 +M +R +C +c5 +c40 +c5 +c40 +c5 +c40 +c5 +c40 +Up-Down [4] +36.9 +68.5 +27.6 +36.7 +57.1 +72.4 +117.9 +120.5 +SGAE [49] +37.8 +68.7 +28.1 +37.0 +58.2 +73.1 +122.7 +125.5 +ETA [19] +38.9 +70.2 +28.6 +38.0 +58.6 +73.9 +122.1 +124.4 +APN [48] +38.9 +70.2 +28.8 +38.0 +58.7 +73.7 +126.3 +127.6 +NG-SAN [11] +38.8 +70.2 +29.0 +38.4 +58.7 +74.0 +126.3 +128.6 +Dual-Global [45] +39.1 +71.2 +28.9 +38.4 +58.9 +74.4 +126.3 +129.2 +AoANet [14] +39.4 +71.2 +29.1 +38.5 +58.9 +74.5 +126.9 +129.6 +M2 Transformer [8] +39.7 +72.8 +29.4 +39.0 +59.2 +74.8 +129.3 +132.1 +RSTNet [54] +39.7 +72.5 +29.3 +38.7 +59.2 +74.2 +130.1 +132.4 +ACF +39.0 +71.3 +29.2 +39.2 +59.2 +74.2 +130.2 +132.3 +ACF achieves comparable performances with ViTCAP- +large [10] that distills knowledge from Google-CC [37], +SBU Caption dataset [30], MSCOCO [22], and Visual +Genome dataset [17], which uses 9.9M image-text pairs +and 4.1M independent images to pretrain a detector-free IC +model. However, we only use the captions from MSCOCO +to train our ACF. Moreover, compared with APN♯ [48] +which inserts an additional clustering matrix into the Self- +ATT layers into the decoder, ACF achieves higher per- +formance since it inserts the clustering matrix in both vi- +sion encoder and language decoder to build a homogeneous +model. +Also, we submit the single-model results to the online +server for testing, which is shown in Table 5. We can see +that ACF achieves the best performance than the other mod- +els, even we do not ensemble the results as AoANet [14], +M2 Transformer [8], and RSTNet [54]. +Limitations and Potential Solutions. From Table 4, we +can find that PureT-Swin [42] achieves higher CIDEr than +ours. There are two major reasons cause this. Firstly, PureT- +Swin extracts visual features from Swin Transformer [24] +and then still uses Swin Transformer as the visual encoder +to deal with the extracted features. For ACF, the used vision +encoder is quite different from Swin Transformer that they +apply shifted fixed-size windows, while we insert an adap- +tive clustering matrix into the Transformer. In this way, the +whole captioning model (including the vision extractor) is +not a strictly homogeneous structure. Also, it can be seen +that ACF outperforms PureT-standard which applies a stan- +dard Transformer as the vision encoder, which means that +once PureT is not homogeneous, their performance will be +worse. +Secondly, they end-to-end train the whole architecture by +captioning data since Swin Transformer [24] provides well- +trained parameters that PureT does not need to train their +visual extractor from scratch. However, this requires heavy +computation resources to end-to-end train the visual extrac- +tor by image annotations while we now cannot afford such +computation burdens. However, even with these two limi- +tations, it can be found that ACF still achieves comparable +performances compared with PureT. +To solve these limitations, we prepare to extend the com- +putation resources like the GPU servers to build a novel pure +vision global-local Transformer where ACF prior is used +to learn hierarchical structure. And then using this model +to extract visual features for solving more vision-language + +tasks, e.g., by building a homogeneous ACF-based vision- +language model. +5. Conclusion +We propose a novel global-local Transformer named as +Ada-ClustFormer (ACF) that can adaptively cluster the in- +put elements for carrying self-attention (Self-ATT) to learn +global-local contexts. Specifically, this is achieved by in- +serting a clustering matrix into the Self-ATT layer, where +the probability terms are calculated from the input data and +thus ACF can adaptively cluster the elements. Moreover, +we use ACF to build an image captioning model to transfer +more structural commonalities for better captions. The ex- +periment results confirm the effectiveness of the proposed +model. +References +[1] Abhaya Agarwal and Alon Lavie. Meteor: An automatic +metric for mt evaluation with high levels of correlation with +human judgments. Proceedings of WMT-08, 2007. +[2] Mahtab Ahmed, Muhammad Rifayat Samee, and Robert E +Mercer. You only need attention to traverse trees. In Pro- +ceedings of the 57th Annual Meeting of the Association for +Computational Linguistics, pages 316–322, 2019. +[3] Peter Anderson, Basura Fernando, Mark Johnson, and +Stephen Gould. Spice: Semantic propositional image cap- +tion evaluation. In European conference on computer vision, +pages 382–398. Springer, 2016. +[4] Peter Anderson, Xiaodong He, Chris Buehler, Damien +Teney, Mark Johnson, Stephen Gould, and Lei Zhang. +Bottom-up and top-down attention for image captioning and +visual question answering. In Proceedings of the IEEE con- +ference on computer vision and pattern recognition, pages +6077–6086, 2018. +[5] Peter W Battaglia, Jessica B Hamrick, Victor Bapst, Al- +varo Sanchez-Gonzalez, Vinicius Zambaldi, Mateusz Ma- +linowski, Andrea Tacchetti, David Raposo, Adam Santoro, +Ryan Faulkner, et al. Relational inductive biases, deep learn- +ing, and graph networks. arXiv preprint arXiv:1806.01261, +2018. +[6] Iz Beltagy, Matthew E Peters, and Arman Cohan. +Long- +former: The long-document transformer. +arXiv preprint +arXiv:2004.05150, 2020. +[7] Boyu Chen, Peixia Li, Chuming Li, Baopu Li, Lei Bai, Chen +Lin, Ming Sun, Junjie Yan, and Wanli Ouyang. Glit: Neural +architecture search for global and local image transformer. +In Proceedings of the IEEE/CVF International Conference +on Computer Vision, pages 12–21, 2021. +[8] Marcella Cornia, Matteo Stefanini, Lorenzo Baraldi, and +Rita Cucchiara. Meshed-memory transformer for image cap- +tioning. +In Proceedings of the IEEE/CVF Conference on +Computer Vision and Pattern Recognition, pages 10578– +10587, 2020. +[9] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, +Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, +Mostafa Dehghani, Matthias Minderer, Georg Heigold, Syl- +vain Gelly, Jakob Uszkoreit, and Neil Houlsby. An image is +worth 16x16 words: Transformers for image recognition at +scale. ICLR, 2021. +[10] Zhiyuan Fang, Jianfeng Wang, Xiaowei Hu, Lin Liang, Zhe +Gan, Lijuan Wang, Yezhou Yang, and Zicheng Liu. Injecting +semantic concepts into end-to-end image captioning. In Pro- +ceedings of the IEEE/CVF Conference on Computer Vision +and Pattern Recognition, pages 18009–18019, 2022. +[11] Longteng Guo, Jing Liu, Xinxin Zhu, Peng Yao, Shichen +Lu, and Hanqing Lu. Normalized and geometry-aware self- +attention network for image captioning. In Proceedings of +the IEEE/CVF Conference on Computer Vision and Pattern +Recognition, pages 10327–10336, 2020. +[12] Simao Herdade, Armin Kappeler, Kofi Boakye, and Joao +Soares. Image captioning: Transforming objects into words. +In Advances in Neural Information Processing Systems, +pages 11137–11147, 2019. +[13] Xiaowei Hu, Zhe Gan, Jianfeng Wang, Zhengyuan Yang, +Zicheng Liu, Yumao Lu, and Lijuan Wang. +Scaling up +vision-language pre-training for image captioning. In Pro- +ceedings of the IEEE/CVF Conference on Computer Vision +and Pattern Recognition, pages 17980–17989, 2022. +[14] Lun Huang, Wenmin Wang, Jie Chen, and Xiao-Yong Wei. +Attention on attention for image captioning. In Proceedings +of the IEEE International Conference on Computer Vision, +pages 4634–4643, 2019. +[15] Huaizu Jiang, Ishan Misra, Marcus Rohrbach, Erik Learned- +Miller, and Xinlei Chen. In defense of grid features for visual +question answering. In Proceedings of the IEEE/CVF Con- +ference on Computer Vision and Pattern Recognition, pages +10267–10276, 2020. +[16] Andrej Karpathy and Li Fei-Fei. Deep visual-semantic align- +ments for generating image descriptions. In Proceedings of +the IEEE conference on computer vision and pattern recog- +nition, pages 3128–3137, 2015. +[17] Ranjay Krishna, Yuke Zhu, Oliver Groth, Justin Johnson, +Kenji Hata, Joshua Kravitz, Stephanie Chen, Yannis Kalan- +tidis, Li-Jia Li, David A Shamma, et al. +Visual genome: +Connecting language and vision using crowdsourced dense +image annotations. International Journal of Computer Vi- +sion, 123(1):32–73, 2017. +[18] Hwanhee Lee, +Seunghyun Yoon, +Franck Dernoncourt, +Doo Soon Kim, Trung Bui, and Kyomin Jung. Vilbertscore: +Evaluating image caption using vision-and-language bert. In +Proceedings of the First Workshop on Evaluation and Com- +parison of NLP Systems, pages 34–39, 2020. +[19] Guang Li, Linchao Zhu, Ping Liu, and Yi Yang. +Entan- +gled transformer for image captioning. In Proceedings of +the IEEE/CVF International Conference on Computer Vision +(ICCV), October 2019. +[20] Jinpeng Li, Yichao Yan, Shengcai Liao, Xiaokang Yang, and +Ling Shao. +Local-to-global self-attention in vision trans- +formers. arXiv preprint arXiv:2107.04735, 2021. +[21] Xiujun Li, Xi Yin, Chunyuan Li, Pengchuan Zhang, Xiaowei +Hu, Lei Zhang, Lijuan Wang, Houdong Hu, Li Dong, Furu +Wei, et al. Oscar: Object-semantics aligned pre-training for + +vision-language tasks. In European Conference on Computer +Vision, pages 121–137. Springer, 2020. +[22] Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, +Pietro Perona, Deva Ramanan, Piotr Doll´ar, and C Lawrence +Zitnick. Microsoft coco: Common objects in context. In +European conference on computer vision, pages 740–755. +Springer, 2014. +[23] Wei Liu, Sihan Chen, Longteng Guo, Xinxin Zhu, and Jing +Liu. Cptr: Full transformer network for image captioning. +arXiv preprint arXiv:2101.10804, 2021. +[24] Ze Liu, Yutong Lin, Yue Cao, Han Hu, Yixuan Wei, Zheng +Zhang, Stephen Lin, and Baining Guo. Swin transformer: +Hierarchical vision transformer using shifted windows. In +Proceedings of the IEEE/CVF International Conference on +Computer Vision, pages 10012–10022, 2021. +[25] Yunpeng Luo, Jiayi Ji, Xiaoshuai Sun, Liujuan Cao, +Yongjian Wu, Feiyue Huang, Chia-Wen Lin, and Rongrong +Ji. Dual-level collaborative transformer for image caption- +ing. +In Proceedings of the AAAI Conference on Artificial +Intelligence, volume 35, pages 2286–2293, 2021. +[26] Minh-Thang Luong, Hieu Pham, and Christopher D Man- +ning. Effective approaches to attention-based neural machine +translation. arXiv preprint arXiv:1508.04025, 2015. +[27] Ron Mokady, Amir Hertz, and Amit H Bermano. +Clip- +cap: +Clip prefix for image captioning. +arXiv preprint +arXiv:2111.09734, 2021. +[28] Van-Quang Nguyen, Masanori Suganuma, and Takayuki +Okatani. +Grit: +Faster and better image captioning +transformer using dual visual features. +arXiv preprint +arXiv:2207.09666, 2022. +[29] Xuan-Phi Nguyen, Shafiq Joty, Steven Hoi, and Richard +Socher. Tree-structured attention with hierarchical accumu- +lation. In International Conference on Learning Representa- +tions, 2020. +[30] Vicente Ordonez, +Girish Kulkarni, +and Tamara Berg. +Im2text: Describing images using 1 million captioned pho- +tographs. Advances in neural information processing sys- +tems, 24, 2011. +[31] Yingwei Pan, Ting Yao, Yehao Li, and Tao Mei. X-linear +attention networks for image captioning. In CVPR, pages +10971–10980, 2020. +[32] Kishore Papineni, Salim Roukos, Todd Ward, and Wei-Jing +Zhu. Bleu: a method for automatic evaluation of machine +translation. In Proceedings of the 40th annual meeting of the +Association for Computational Linguistics, pages 311–318, +2002. +[33] Samrudhdhi B Rangrej, Kevin J Liang, Tal Hassner, and +James J Clark. Glitr: Glimpse transformers with spatiotem- +poral consistency for online action prediction. arXiv preprint +arXiv:2210.13605, 2022. +[34] S Ren, K He, R Girshick, and J Sun. Towards real-time ob- +ject detection with region proposal networks. Advances in +neural information processing systems, 2015. +[35] Steven J Rennie, Etienne Marcheret, Youssef Mroueh, Jerret +Ross, and Vaibhava Goel. Self-critical sequence training for +image captioning. In Proceedings of the IEEE conference on +computer vision and pattern recognition, pages 7008–7024, +2017. +[36] Lin CY ROUGE. +A package for automatic evaluation of +summaries. +In Proceedings of Workshop on Text Summa- +rization of ACL, Spain, 2004. +[37] Piyush Sharma, Nan Ding, Sebastian Goodman, and Radu +Soricut. Conceptual captions: A cleaned, hypernymed, im- +age alt-text dataset for automatic image captioning. In Pro- +ceedings of the 56th Annual Meeting of the Association for +Computational Linguistics (Volume 1: Long Papers), pages +2556–2565, 2018. +[38] Ying Hua Tan and Chee Seng Chan. Phrase-based image +caption generator with hierarchical lstm network. Neurocom- +puting, 333:86–100, 2019. +[39] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszko- +reit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia +Polosukhin. Attention is all you need. Advances in neural +information processing systems, 30, 2017. +[40] Ramakrishna Vedantam, C Lawrence Zitnick, and Devi +Parikh. Cider: Consensus-based image description evalua- +tion. In Proceedings of the IEEE conference on computer +vision and pattern recognition, pages 4566–4575, 2015. +[41] Oriol Vinyals, Alexander Toshev, Samy Bengio, and Du- +mitru Erhan. Show and tell: A neural image caption gen- +erator. In Proceedings of the IEEE conference on computer +vision and pattern recognition, pages 3156–3164, 2015. +[42] Yiyu Wang, Jungang Xu, and Yingfei Sun. End-to-end trans- +former based model for image captioning. In Proceedings of +the AAAI Conference on Artificial Intelligence, pages 2585– +2594, Jun. 2022. +[43] Yau-Shian Wang, Hung-Yi Lee, and Yun-Nung Chen. Tree +transformer: Integrating tree structures into self-attention. +arXiv preprint arXiv:1909.06639, 2019. +[44] Chuhan Wu, Fangzhao Wu, Tao Qi, and Yongfeng Huang. +Hi-transformer: hierarchical interactive transformer for effi- +cient and effective long document modeling. arXiv preprint +arXiv:2106.01040, 2021. +[45] Tiantao Xian, Zhixin Li, Canlong Zhang, and Huifang Ma. +Dual global enhanced transformer for image captioning. +Neural Networks, 148:129–141, 2022. +[46] Kelvin Xu, Jimmy Ba, Ryan Kiros, Kyunghyun Cho, Aaron +Courville, Ruslan Salakhudinov, Rich Zemel, and Yoshua +Bengio. Show, attend and tell: Neural image caption gen- +eration with visual attention. In International conference on +machine learning, pages 2048–2057. PMLR, 2015. +[47] Jianwei Yang, Chunyuan Li, Pengchuan Zhang, Xiyang Dai, +Bin Xiao, Lu Yuan, and Jianfeng Gao. Focal attention for +long-range interactions in vision transformers. Advances in +Neural Information Processing Systems, 34:30008–30022, +2021. +[48] Xu Yang, Chongyang Gao, Hanwang Zhang, and Jianfei Cai. +Auto-parsing network for image captioning and visual ques- +tion answering. In Proceedings of the IEEE/CVF Interna- +tional Conference on Computer Vision, pages 2197–2207, +2021. +[49] Xu Yang, Kaihua Tang, Hanwang Zhang, and Jianfei Cai. +Auto-encoding scene graphs for image captioning. In Pro- +ceedings of the IEEE/CVF Conference on Computer Vision +and Pattern Recognition, pages 10685–10694, 2019. + +[50] Xu Yang, Hanwang Zhang, Guojun Qi, and Jianfei Cai. +Causal attention for vision-language tasks. In Proceedings +of the IEEE/CVF Conference on Computer Vision and Pat- +tern Recognition, pages 9847–9857, 2021. +[51] Ting Yao, Yingwei Pan, Yehao Li, and Tao Mei. +Hierar- +chy parsing for image captioning. +In Proceedings of the +IEEE/CVF International Conference on Computer Vision, +pages 2621–2629, 2019. +[52] Pengchuan Zhang, Xiujun Li, Xiaowei Hu, Jianwei Yang, +Lei Zhang, Lijuan Wang, Yejin Choi, and Jianfeng Gao. +Vinvl: Revisiting visual representations in vision-language +models. +In Proceedings of the IEEE/CVF Conference on +Computer Vision and Pattern Recognition, pages 5579– +5588, 2021. +[53] Hengshuang Zhao, Jiaya Jia, and Vladlen Koltun. Explor- +ing self-attention for image recognition. In Proceedings of +the IEEE/CVF Conference on Computer Vision and Pattern +Recognition, pages 10076–10085, 2020. +[54] Luowei Zhou, Hamid Palangi, Lei Zhang, Houdong Hu, Ja- +son Corso, and Jianfeng Gao. Unified vision-language pre- +training for image captioning and vqa. In Proceedings of +the AAAI Conference on Artificial Intelligence, volume 34, +pages 13041–13049, 2020. + diff --git a/AtAzT4oBgHgl3EQf__8r/content/tmp_files/load_file.txt b/AtAzT4oBgHgl3EQf__8r/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..caf61908230a277094cf8fdcb56669cd932946c8 --- /dev/null +++ b/AtAzT4oBgHgl3EQf__8r/content/tmp_files/load_file.txt @@ -0,0 +1,944 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf,len=943 +page_content='Adaptively Clustering Neighbor Elements for Image Captioning Zihua Wang1,2 Xu Yang1 Haiyang Xu2* Hanwang Zhang3 Chenliang Li2 Songfang Huang2 Fei Huang2 Yu Zhang1* 1 School of Computer Science & Engineering, Key Lab of Computer Network & Information Integration (Ministry of Education), Southeast Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', Nanjing, China 2Alibaba Group 3 School of Computer Science & Engineering, Nanyang Technological Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' {zihua, 101013120, zhang yu}@seu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='cn,{shuofeng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='xhy, lcl193798, songfang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='hsf, f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='huang}@alibaba-inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='com, hanwangzhang@ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='sg Abstract We design a novel global-local Transformer named Ada- ClustFormer (ACF) to generate captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' We use this name since each layer of ACF can adaptively cluster input el- ements to carry self-attention (Self-ATT) for learning lo- cal context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Compared with other global-local Transform- ers which carry Self-ATT in fixed-size windows, ACF can capture varying graininess, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', an object may cover dif- ferent numbers of grids or a phrase may contain diverse numbers of words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' To build ACF, we insert a probabilis- tic matrix C into the Self-ATT layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' For an input se- quence {s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', sN}, Ci,j softly determines whether the sub-sequence {si, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', sj} should be clustered for carrying Self-ATT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' For implementation, Ci,j is calculated from the contexts of {si, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', sj}, thus ACF can exploit the input itself to decide which local contexts should be learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' By us- ing ACF to build the vision encoder and language decoder, the captioning model can automatically discover the hid- den structures in both vision and language, which encour- ages the model to learn a unified structural space for trans- ferring more structural commonalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' The experiment re- sults demonstrate the effectiveness of ACF that we achieve CIDEr of 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8, which outperforms most SOTA captioning models and achieve comparable scores compared with some BERT-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' The code will be available in the sup- plementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Introduction Image Captioning (IC) aims to learn a shared vision- language representation space for facilitating the transfer of multimodal knowledge to generate visually grounded sen- Corresponding authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' tence [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Two prevailing deep learning techniques help the IC model learn such space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' The first one is the vi- sion encoder-language decoder pipeline [41] which back- propagates the language semantic to the visual encoder and another one is the attention mechanism [46] which di- rectly bridges between vision and language domains for transferring multimodal knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Transformers [39], which build the encoder and decoder based on dense at- tention operations, have both of the above-mentioned ad- vantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Transformers have two types of attention opera- tions which are self-attention (Self-ATT) and cross-modal attention (Cross-ATT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' From the perspective of structure learning, Self-ATT applies the fully connected (FC) graph prior to the data sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' By using Self-ATT in both encoder and decoder, the graph structures of both vision and language data can be discovered and Cross-ATT helps transfer these structural commonalities for narrowing the modality gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Therefore, Transformer prevails in IC tasks [10,12,13,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Interestingly, structure learning is one of the most sig- nificant research directions of IC since the paired vision- language data usually share a unified internal semantic structure although they have diverse external appearances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Thus, if this unified semantic structure is captured, more structural commonalities can be transferred for generating better captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Motivated by this, various IC models are proposed to exploit scene graphs [5, 21, 49] or hierarchy trees [43, 51] to narrow the domain gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' However, such structures need additional well-trained parsers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Moreover, vision and language parsers usually have domain gaps that the parsed structures of the paired image-sentence may not match, which may even weaken the effectiveness of these IC models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' We prefer an IC model that can adaptively dis- cover the unified semantic structures to remove the costs of the additional structure annotations and more importantly, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='01955v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='CV] 5 Jan 2023 (a) Fixed-Size Transformer s1 s2 s3 s4 s5 s6 s7 s8 Input 1-st layer 2-nd layer 3-rd layer (b) ACF s1 s2 s3 s4 s5 s6 s7 s8 Input 1-st layer 2-nd layer 3-rd layer … … … … (c) ACF-based IC riding a snow board on snow A man riding a snow board on snow A man riding a snow board on snow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' riding a A man snow board on snow A man Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' (a) Transformer with fixed-size windows (size = 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' (b) ACF which adjusts the window size according to the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' (c) ACF-based IC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' The left/right part shows how the vision/language ACFs cluster image grids/language words for transferring struc- tural commonalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' to learn a unified structure space for transferring structural commonalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Transformer seems to be a good starting point since it can implicitly build graphs by Self-ATT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' However, it exploits the FC graph prior, while the useful semantic structure is usually sparse and hierarchical like the scene graphs or trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' To discover more sparse structures, re- searchers design various global-local Transformers [20,29, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' As sketched in Figure 1(a), these Transformers grad- ually merge the neighbor elements in fixed-size windows into bigger clusters and carry Self-ATT in each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' For example, the 1-st layer clusters 2 neighboring elements like {s1, s2} to carry Self-ATT for local contexts and the 2- nd layer merges {s1, s2} and {s3, s4} into a bigger one to learn more global context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Then a hierarchical struc- ture is built from lower to higher layers where local and global contexts are respectively captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' However, these Transformers still do not satisfy our requirement since vi- sion and language data have diverse graininess, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', objects may cover varying grids and phrases may compose different numbers of words, while fixed-size windows cannot effec- tively capture such varying graininess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' To capture the varying graininess, we propose to Adaptively Cluster the neighbor elements to carry Self-ATT and named the novel Transformer as Ada- ClustFormer (ACF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' As shown in Figure 1(b), in each layer, the window size is not fixed but can be adjusted to each specific input sequence, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', in the 1-st layer, {s1, s2, s3}, {s4}, {s5, s6}, {s7}, {s8} are respectively clustered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' The higher layers merge small clusters into big- ger ones for global contexts, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', the 2-nd layer respectively merges {s1, s2, s3, s4, s5, s6}, {s7, s8} into two clusters to carry Self-ATT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' To achieve this adaptive clustering, we in- sert a probabilistic clustering matrix C into the Self-ATT layer, where the probability Cij softly determines whether the sub-sequence {si, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', sj} should be clustered or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' To calculate Cij, we consider whether the next element sj is similar to the mean-pooling of {si, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', sj−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Thus ACF can adjust the window of Self-ATT based on each specific data sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' To construct an IC model based on ACF, besides build- ing 1-D ACF for the language decoder, we also extend it to the 2-D ACF as the vision encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In this way, both the visual encoder and language decoder can automatically discover the hidden structures of the image and language data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' This means that the ACF model does not need any additional structure annotations as some previous IC mod- els [2, 5] but still exploits the sparse structures implied in both vision and language data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' For example, as shown in Figure 1(c), a visual ACF can merge the smaller grids into bigger regions to capture both grid-level [15] and region- level [4] contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' And the language one gradually clus- ters the single words into phrases to generate the captions in an imaginary phrase-by-phrase manner [38, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' More importantly, compared with certain global-local Transform- ers which are exclusively developed in vision and language domains [24, 47], the visual and language ACF exploit the same way to discover hidden structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' So, our ACF model is a homogeneous structure that helps transfer more struc- tural commonalities between vision and language domains, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', as shown in Figure 1(c), the patches of the object “snow board” is clustered in the image and correspondingly, the phrase “a snow board” is also clustered in the language do- main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In summary, our contributions can be listed as follows: We propose ACF that can adaptively capture varying graininess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' We extend ACF to the 2-D case for building a homo- geneous IC model that learns unified structural space for transferring more structural commonalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' The experimental results show that our ACF model outperforms the classic Transformers in IC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Related Work Image Captioning (IC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' IC aims to generate descriptions according to the given images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Typically, an encoder- decoder paradigm is used to convert visual inputs to se- quence outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In the early stage, image features are ex- tracted by CNN-based encoders, as the input of the RNN- based decoders [4, 16, 35, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' For example, Up-Down [4] employs a Faster R-CNN [34] to extract image region fea- tures and LSTM networks to generate sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Nowadays, Transformer-based models have shown their might in Neural Language Process (NLP) and replace RNN- based decoders in IC [12, 14, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Subsequently, more ad- vanced Transformer-based decoders are proposed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', M2 Transformer [8] proposes a meshed-memory Transformer to interact with the low-level and high-level features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' X- Linear Transformer [31] selectively capitalizes the visual information from image regions by bilinear pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' However, these models still use CNN-based feature ex- tractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' More recently, witnessing the boom of Vision Transformers (ViT) [9, 24], researchers use ViT-based vi- sual encoders for captioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' For instance, CPTR [23] in- troduces grid-based features that are extracted by ViT [9] instead of using the ROI-based features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' DLCT [25] fuses the ROI-based features with the grid-based features to over- come the shortcoming of both features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Besides that, some models exploit the knowledge distilled from Vision- Language BERTs for better captions [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' VinVL [52] and GRIT [28] propose the object detection model in IC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' ClipCAP [27] and LEMON [13] introduce large-scale pre- training on IC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Noteworthy, the methods above employ the ViT [9] or Swin Transformer [24] as their backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Thus, our ACF adopts the Swin Transformer as our encoder back- bone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Among the previous IC models, Auto-Parsing Network (APN) [48] has a similar motivation as ours, which also in- serts a clustering matrix into the Self-ATT layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' However, Ada-ClustFormer (ACF) calculates this matrix differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' APN only considers whether pairwise neighboring elements should be clustered or not, while we calculate this proba- bility from a more global scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Specifically, we consider whether the next element is similar to the previous clustered elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' More importantly, we extend our ACF into the 2- D case, which can adaptively cluster the visual patches into regions, while APN only treats a sequence of ROI features as the visual input and still applies a 1-D clustering matrix to address it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' More comparisons will be given in the supple- mentary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Global-Local Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' To alleviate the fully con- nected graph prior in Transformer, researchers propose var- ious global-local Transformers to learn sparse structures of the language [6, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' For example, Global-local [26] intro- duces a fixed-size of the global and local attention model in neural machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Longformer [6] proposes global and local window attentions, which can provide inductive bias and long sequence representation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Hi- Transformer [44] learns sentence-level and document-level semantics through the hierarchical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' The global-local Transformer mechanism is also effec- tive in vision area [7, 25, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Pairwise and patchwise self- attention are proposed in image recognition [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Further- more, GLiT [7] proposes to adaptively trade off the global and local information of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' DLCT [25] explores the global and local information by the combination of grid- based features and ROI-based features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' However, these models are exclusively developed in a single domain (either NLP or CV), while our ACF provides a general approach in both the vision and language domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Thus, using ACF to build the IC model encourages learn- ing a unified structure space for transferring more structure commonalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Ada-ClustFormer IC model Compared with the classic Transformer, Ada- ClustFormer (ACF) inserts an adaptively clustering matrix C into each self-attention (Self-ATT) layer to adaptively control the scope of Self-ATT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' The calculation of C is detailed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 where we first show the 1-D language case and then extend it to the 2-D vision case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' By stacking these revised Self-ATT layers, ACF can be built for constructing the vision encoder and language decoder for captioning (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Ada-ClustFormer Multi-Head Attention (MHA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' ACF is built based on Transformer, whose most elemental building block is the Multi-Head Attention (MHA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Given the query Q ∈ RNQ×d, key K ∈ RNK×d, and value V ∈ RNV ×d, MHA calculates the output Z = MHA(Q, K, V) as: Input: Q, K, V ATT: Al = Softmax(QWQ l (KWK l )T √ d ) Head : Hl = AlVWV l , Multi-Head: H = [H1, H2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', Hh]WH, Output: Z = LN(H + Q), (1) where WQ l , WK l , WV l ∈ Rd×dh, WH l ∈ Rd×d are all learn- able parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' h denotes the head number and dh = d/h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Al is the l-th attention matrix corresponding to the l-th head Hl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [·] is the concatenation operation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' and LN denotes to the Layer Normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Given an input sequence S = {s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', sN}, if Q = K = V = S, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' (1) is also called self-attention (Self- ATT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Self-ATT captures the global contexts between any two elements si and sj by calculating the pairwise atten- tion weight in the “ATT” operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' From the perspective of structure learning [5], single-head Self-ATT constructs a fully-connected (FC) graph where the nodes are the ele- ments of S and the pairwise edges are weighted by the pair- wise attention weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Correspondingly, a h-head Self-ATT constructs h FC graphs with different edge weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Adaptive Clustering Matrix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' To sparsify this FC-graph, researchers [9, 24] propose to carry Self-ATT in fixed-size windows, which is achieved by revising “Head” in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' (1): C-based Head : H = Softmax(A ⊗ C)VWV , (2) where “⊗” denotes the element-wise production;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' C is a N × N binary clustering matrix that only the elements in the window can attend to each other, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', if the win- dow size is w, Ci,j = 1 if |i − j| ≤ w and Ci,j = 0 if |i − j| > w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' However, language or vision data usually have diverse graininess, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', a phrase may contain different numbers of words or an object may cover diverse spatial regions, while the fixed-size windows can not capture the varying graininess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' To amend this, we revise the binary C to a probabilistic one where Ci,j softly determines whether to cluster the em- beddings from si to sj for carrying Self-ATT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Then if Ci,j is small, the pairwise attention in A between si and sj is weakened in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' (2), which means si and sj are less likely to stay in the same cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' To adaptively decide the win- dow size according to each specific input for capturing the varying graininess, we use the input itself to calculate Ci,j: Ci,j = P(si, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', sj) = j� k=i P(sk|si, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', sk−1), (3) where the joint distribution is decomposed to the produc- tions of conditional distributions P(sk|si, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', sk−1), which softly decides whether to merge a new element sk into the sub-sequence {si, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', sk−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In the implementation, P(sk|si, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', sk−1) is calculated as: P(sk|si, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', sk−1) = Sigmoid(FC([sk, si:k−1])), (4) where si:k−1 is the mean pooling of the embeddings from si to sk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Intuitively, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' (4) exploits the context of the whole sub-sequence {si, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', sk−1} to decide whether to merge a new element {sk} into this sub-sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' (3) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' (4) only make sense when i < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Since clustering the embeddings from si to sk equals to cluster- ing from sk to si, we set Ci,k = Ck,i if i > k and since a single element si is itself a cluster, we set Ci,i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' (3), we can also find that: Ci,j =P(sj|si, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', sj−1) × P(si, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', sj−1) =P(sj|si, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', sj−1) × Ci,j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' (5) Since P(sj|si, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', sj−1) ≤ 1, we have Ci,j ≤ Ci,j−1, which means that two elements in the shorter distance are more likely to be clustered for carrying Self-ATT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In this way, local contexts are encouraged to be captured, as is shown in Figure 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Stacking Revised Self-ATT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' To learn global contexts, we can stack these revised Self-ATT layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' When stacking, we hope that the higher layers will carry Self-ATT in bigger windows than the lower layers to capture the global con- texts [43, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' To achieve this, for the m-th layer, we re- calculate C(m) as ˜C(m): ˜C(m) = (1 − C(m)) ˜C(m−1) + C(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' (6) s1 s2 s3 s4 s5 s6 s1 s2 s3 s4 s5 s6 C1,4 = C1,3 × P( s4 | s1, s2, s3) Sigmoid(FC([s4, s1:s3])) (a) Calculation of C1,4 (b) C(2) ≥ C(1) 1-st layer 2-nd layer ~ ~ Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' (a) shows how to calculate C1,4, where the shade denotes the probability value, the darker the color, the larger the probability value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' (b) shows that the clustered elements in the lower layer will be further clustered in a higher layer, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', the color of {s1, s2, s3} in the 2-nd layer is darker than the 1-st layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Horizontal Upsampling (a) Calculation of C1,4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1,3 (b) Down-up Sampling Strategy Ph(v1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', v4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1) Pv(v1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', v1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3) s1 s2 s4 s3 s1 s2 C1,2 C2,3 ∏ ∏ C1,4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1,3 Horizontal Upsampling (a) Calculation of C1,4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1,3 (b) Down-up Sampling Strategy Ph(v1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', v4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1) Pv(v1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', v1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3) s1 s2 s4 s3 s1 s2 C1,2 C2,3 ∏ ∏ C1,4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1,3 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' (a) The example of 2-D C, where C1,4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1,3 is used as the example, which is decomposed into vertical and horizontal di- rections probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' (b) Overview of the Down-Up Sampling Strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Then ˜C(m) is used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' (2) when m > 1 and ˜C(1) = C(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Since 0 ≤ C(m) i,j ≤ 1, ˜C(m) i,j is a convex combination of ˜C(m−1) i,j and 1, which means that ˜C(m−1) i,j ≤ ˜C(m) i,j ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' If ˜C(m−1) i,j is large, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', the sub-sequence {si, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', sj} should be clustered in the (m − 1)-th layer, then ˜C(m) i,j must be larger, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', {si, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', sj} is also clustered in the m-th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' For example, Figure 2(b) shows that the 2-nd layer will further cluster {s1, s2, s3} since ˜C(1) 1,3 ≤ ˜C(2) 1,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Thus, the higher layers will carry Self-ATT in a bigger window than the lower layers to learn more global contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' 2-D Clustering Matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' (3) shows how to calculate C when the input is a 1-D language sequence, next we extend it to the 2-D vision surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Given a 2-D fea- ture map V = {v1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', vH,W }, we use Ci,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='x,y to de- note the probability that softly decides whether a sub-region {vi,x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', vj,y} should be clustered or not, which is: Ci,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='x,y = P(vi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', vj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='y) = j � k=i y � u=x P(vk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='u|vi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='x, vi+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', vk−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='u−1) (7) where i, j and x, y respectively denote the horizontal and vertical dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' To cover all the sub-regions in a H×W Image Self-ATT Add&LN 1-D C Self-ATT Add&LN Words Cross-ATT Add&LN Captioning: Z me× Encoder Decoder md× Q,K,V Q,K,V K,V Q 2-D C Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Overview of our ACF-based encoder-decoder IC model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' The “Add&LN” is the Add and Layer Normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' me/md rep- resent the number of the encoder/decoder layers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' map, it requires applying O(H2 × W 2) times for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' (4) to get all the probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' To reduce the computation burden, we apply the independence assumption to decompose the 2-D distribution into two independent ones, which respec- tively correspond to the horizontal and vertical dimensions: P(vi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', vj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='y) = Ph(vi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='vj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='x)Pv(vi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', vi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='y) = j � k=i Ph(vk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='x|vi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', vk−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='x) y � u=x Pv(vi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='x|vi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', vi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='u−1), (8) In this way, we only need to apply O(H2 + W 2) times for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' (4) and once matrix production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Noteworthy, as sketched in Figure 2, for the 2-D region which spans the horizontal axis from i to j and the vertical axis from x to y, we use the left-most vertical and top-most hor- izontal to calculate two 1-D distributions and then mul- tiply them to get Ci,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='x,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' As Figure 3(a) shows, to calculate C1,4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1,3, for the vertical distribution Pv, the horizontal ordinate is fixed to 1 and the vertical or- dinate changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Ph(vk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1|v1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', vk−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1)|k=1,2,3,4 and Pv(v1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='u|v1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', v1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='u−1)|u=1,2,3 are calculated in the same way as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' The above-mentioned symmetric character- istic is also applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Down-Up Sampling Strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' If the sequence (feature map) is too long (big), we can apply the Down-Up Sam- pling Strategy to reduce the computation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' We use a 1-D language case as an example to show this strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' For S = {s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', sL}, we can downsample it to ¯S = {¯s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', ¯sL/2} where ¯si is the mean pooling of s2∗i−1 and s2∗i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Then ¯S is used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' (3) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' (4) to get ¯ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' To upsample ¯C to the original size, we set Ci,j = ¯ C⌈i/2⌉,⌈j/2⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Figure 3(b) shows one simple case where L = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Encoder-Decoder Architecture As is shown in Figure 4, we apply the ACF to build the vision encoder and language decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Compared to the clas- sic Transformer, our ACF introduces clustering-restrained attention head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Specifically, in encoder, we calculate a 2-D clustering matrix C (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' (7)) to softly cluster the ele- ments for carrying Self-ATT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Similarly, in decoder, the at- tention head is revised with the 1-D C (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' (5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' The output of this encoder-decoder is used to calculate the word distributions Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' To train our IC model, we optimize the model by min- imizing the cross-entropy loss and maximizing the Rein- forcement learning (RL) [35] reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' First, we train the model by minimizing the cross-entropy loss: LCE = − log P(Z∗), (9) where Z∗ is the ground-truth captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Then, we further train the model by minimizing the negative reward: Lrl = −EZs∼P (Z)(S(Z∗, Zs)), (10) where Zs is sampled from Z, E represents the mathemat- ical expectation, and S represents the evaluation metrics, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', CIDEr [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Dataset, Metrics, and Settings MSCOCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Following [8, 12, 14, 31, 48], we train and evaluate our model on MSCOCO [22], which contains 123, 287 images, and each one is annotated with 5 cap- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In the experiments, we use the Karpathy split (113,287/5,000/5,000 train/val/test images) [16] for offline training and the official split (40775 test images) for online testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' We adopt five widely-used metrics in caption- ing for evaluation, including BLEU [32], METOR [1], ROUGE-L [36], CIDEr [40], and SPICE [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In the training process, we convert all the captions into lowercase and delete all the words that occur less than 6 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' The remaining 9487 words are regarded as our vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' We adopt Swin Transformer [24] as the visual encoder to extract the visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' The size of the feature map is H × W = 12 × 12, and we apply the Down-Up Sampling Strategy (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' We train 20/25 epochs in the cross-entropy/RL stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In the cross-entropy stage, the Adam optimizer is used with the learning rate of 5 × 10−5 and decays by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 per 5 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In the RL stage, the learning rate is initialized to 5 × 10−6 and we implement the same decay policy for 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Then the “Reduce- On-Plateau” strategy is applied with a decay rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='5 and patience of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' The batch size is 40 at the whole training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Comparison between with and without Ada-ClustFormer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Models me md B@4 M R C S BASE 6S 6S 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='0 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='7 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='6 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 ACF 1 6C 6S 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='6 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='6 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='7 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='5 ACF 2 6S 6C 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='9 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='7 ACF 6C 6C 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Ablation Studies We conduct extensive ablations for quantifying the dif- ference between classic self-attention (Self-ATT) layers and Ada-ClustFormer (ACF) layers (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1), the im- pact of the depth of the ACF layers (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2), and the impact of the orders of ACF and the Self-ATT layers (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 Differences Between ACF and Self-ATT Comparing Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' To evaluate the effectiveness of the ACF, we ablate our ACF with the following baselines: BASE: We employ 6 Self-ATT encoder layers and de- coder layers, which is shown in Table 1 as “6S”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' ACF 1 / ACF 2: We replace the encoder/decoder with our ACF, which is represented as “6C”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' The results of the ablation are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Compared with BASE, we can find that only using ACF encoder (ACF 1) or decoder (ACF 2) has marginal im- provements, which is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='7 on CIDEr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' However, when combining the ACF encoder and decoder to build a homo- geneous architecture ACF, the improvement is substantial, which is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' This comparison suggests that a homogeneous model transfers more structural commonalities for better captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 Impact of the Layer Depth Comparing Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' ACF 3: We reduce the depth of the encoder and decoder layer to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' ACF 4/ACF 5: The num- ber of the encoder/decoder layers is set to 3 and the number of the decoder/encoder layer remains 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' From Table 2, we observe that stacking 6 layers generally outperforms the 3-layer case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Our method with 6 ACF layers in the encoder and decoder achieves the best performance among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' We also further explore the in- fluence of me by fixing md = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' We present the impact of the number of the encoder layers me in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' It sug- gests that CIDEr approximately linearly increases when me increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3 Impact of the Layer Order Comparing Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' We discuss the combination of the ACF layers and the Self-ATT layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' We freeze the depth Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' The performances with different layer depth Models me md B@4 M R C S ACF 3 3C 3C 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='9 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='0 ACF 4 6C 3C 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='9 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='9 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='7 ACF 5 3C 6C 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='7 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='0 ACF 6C 6C 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' The impact of the layer orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Models me md B@4 M R C S ACF 5 3C 6C 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='7 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='0 ACF 6 3C+ 3S 6C 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='7 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='7 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='9 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='7 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 ACF 7 3S+ 3C 6C 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='5 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='9 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='9 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='9 ACF 2 6S 6C 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='9 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='7 ACF 6C 6C 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 of the decoder layer md = 6 and quantify the influence of the order of the encoders: ACF 5: It stacks 3 ACF lay- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' ACF 6/ACF 7: Both of them have 3 ACF layers and 3 Self-ATT layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' The difference between them is that ACF 7 encodes on 3 Self-ATT layers firstly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' The results are listed in Table 3, where we can see that the performances are not sensitive to the orders of ACF and Self-ATT layers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', ACF 6 and ACF 7 differ only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' We can also find that replacing all the Self-ATT layers with our ACF layers will achieve the best captioning quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' 3 4 5 6 Number of encoder layers 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='0 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='5 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='0 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='5 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='0 CIDEr 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='97 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='6 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='5 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='83 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Impact of the number of encoder layers me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Qualitative Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' We visualize the hierarchical struc- tures of the image and the generated captions in Figure 6 according to the 2-D and 1-D clustering matrix calculated from the 1-st, 3-rd, 5-th, and 6-th layers in encoder and de- coder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' By inspecting the images and captions, we can find that the patches and the words are respectively clustered, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', in the left part of (b), the patches in the “motorcycles” region are clustered, and in the right part, the words “sit- ting on motorcycles” are clustered into a phrase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' More im- portantly, when uniting the image and caption, we can find that structural commonalities are transferred, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', in (b), the “motorcycle” region helps generate the phrase “sitting on motorcycles”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' A woman standing on the door of a train with a suitcase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' a woman standing on the door of a train with a suitcase standing on a woman the door of a a train with suitcase Ground-truth: A woman in white and black dress with suitcase on train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' BASE: A woman standing with a suitcase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' ACF: A woman standing on the door of a train with a suitcase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Two people sitting on motorcycles next to a stop sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' sitting on Two people motorcycles a next to stop sign sitting on motorcycles next to a stop sign Ground-truth: Two people riding motorcycles on a city street.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' BASE: Two people riding black motorcycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' ACF: Two people sitting on motorcycles next to a stop sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Ground-truth: A man with a hat and eye glasses holding a cell phone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' BASE: A man with a cowboy hat holding a cell phone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' ACF: A man wearing a cowboy hat taking a picture with a cell phone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' A man wearing a cowboy hat taking a picture with a cell phone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' wearing cowboy hat a taking cell picture with wearing a cowboy hat a man taking a picture a man a a phone with a cell phone Two people (b) (c) (a) taking a picture with a cell phone A man wearing a cowboy hat Two people sitting on motorcycles next to a stop sign a woman standing on the door of a train with a suitcase Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Examples of the generated captions by BASE and ACF models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' We visualize the 2-D C and 1-D C in the 1-st, 3-rd, 5-th, and 6-th layers as the clustered patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Comparisons with SOTA Comparing Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Nowadays, the SOTA of image cap- tioning has been updated quickly and these models can be categorized into 3 groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' The first one is the meth- ods which use ROI-based features, including Up-Down [4], ORT [12], AoANet [14], M2 Transformer [8], Tree- Transformer [43], APN [48], and X-Transformer [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Among the above methods, Up-Down [4] deploys a famous architecture with a CNN-based encoder and an LSTM- based decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' ORT [12] applies Transformer to lan- guage decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' AoANet [14] and M2 Transformer [8] further improve the attention mechanism on the language decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Tree-Transformer [43] and APN [48] reveal the validity of the utilization of the sequence structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' To capture high-order interaction between sequence and re- gions, X-Transformer [31] introduces a bilinear pooling structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' The second group are the methods using grid- based features: CPTR [23], Dual-Global [45], DLCT [25], and PureT [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Among them, Dual-Global [45] and DLCT [25] combine the grid-based features with the ROI- based features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' PureT [42] end-to-end trains the whole model and PureT-standard/PureT-Swin respectively use Transformer [9]/Swin Transformer [24] as the vision en- coder to deal with the visual features, which is also ex- tracted from a Swin Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' The third group dis- tills the knowledge from large-scale pretraining models: RSTNet [54], and ViTCAP [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Accordingly, we seg- ment the performances into 3 parts in Table 4, where the top/middle/bottom parts are the ROI-based, grid-based, and BERT-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Note that for APN, besides reporting the results in their paper [48], which is got by using ROI- based features, we also report the performances using the same visual features as ours, which is denoted as “APN♯”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' From Table 4, we can see that ACF is com- parable to most of state-of-the-art performance when compared with ROI and grid-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Moreover, STOPS OPSTOPSTOPTable 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' The performances of SOTA methods on MSCOCO Karpathy split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Models Cross-Entroy Loss CIDEr optimization B@4 M R C S B@4 M R C S ROI-based feature Up-Down [4] 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='0 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='7 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='9 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 ORT [12] 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='5 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='0 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='6 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='6 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='7 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='6 AoANet [14] 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='5 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='9 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 M2 Transformer [8] 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='6 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='6 CATT [50] 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='5 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='5 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='9 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='7 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 APN [48] 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='6 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='0 X-Transformer [31] 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='0 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='9 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='7 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='5 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 Grid-based feature CPTR [23] 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='0 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 − APN♯ [48] 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3 Dual-Global [45] 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3 DLCT [25] 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='9 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3 End-to-End training PureT-standard [42] 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='9 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='9 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 PureT-Swin [42] 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='9 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 Visual-language BERT pretraining RSTNet [54] 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='9 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='5 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='6 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3 ViTCAP-small [10] 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='7 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='6 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='0 ViTCAP-large [10] 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='6 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 ACF 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' The scores on the MSCOCO online test server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Models B@4 M R C c5 c40 c5 c40 c5 c40 c5 c40 Up-Down [4] 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='9 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='5 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='6 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='7 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='9 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='5 SGAE [49] 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='7 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='0 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='7 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='5 ETA [19] 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='9 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='6 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='0 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='6 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='9 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 APN [48] 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='9 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='0 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='7 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='7 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='6 NG-SAN [11] 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='0 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='7 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='0 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='6 Dual-Global [45] 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='9 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='9 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 AoANet [14] 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='5 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='9 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='5 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='9 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='6 M2 Transformer [8] 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='7 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='0 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='8 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 RSTNet [54] 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='7 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='5 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='7 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='4 ACF 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='0 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='2 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='3 ACF achieves comparable performances with ViTCAP- large [10] that distills knowledge from Google-CC [37], SBU Caption dataset [30], MSCOCO [22], and Visual Genome dataset [17], which uses 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='9M image-text pairs and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='1M independent images to pretrain a detector-free IC model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' However, we only use the captions from MSCOCO to train our ACF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Moreover, compared with APN♯ [48] which inserts an additional clustering matrix into the Self- ATT layers into the decoder, ACF achieves higher per- formance since it inserts the clustering matrix in both vi- sion encoder and language decoder to build a homogeneous model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Also, we submit the single-model results to the online server for testing, which is shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' We can see that ACF achieves the best performance than the other mod- els, even we do not ensemble the results as AoANet [14], M2 Transformer [8], and RSTNet [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Limitations and Potential Solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' From Table 4, we can find that PureT-Swin [42] achieves higher CIDEr than ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' There are two major reasons cause this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Firstly, PureT- Swin extracts visual features from Swin Transformer [24] and then still uses Swin Transformer as the visual encoder to deal with the extracted features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' For ACF, the used vision encoder is quite different from Swin Transformer that they apply shifted fixed-size windows, while we insert an adap- tive clustering matrix into the Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In this way, the whole captioning model (including the vision extractor) is not a strictly homogeneous structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Also, it can be seen that ACF outperforms PureT-standard which applies a stan- dard Transformer as the vision encoder, which means that once PureT is not homogeneous, their performance will be worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Secondly, they end-to-end train the whole architecture by captioning data since Swin Transformer [24] provides well- trained parameters that PureT does not need to train their visual extractor from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' However, this requires heavy computation resources to end-to-end train the visual extrac- tor by image annotations while we now cannot afford such computation burdens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' However, even with these two limi- tations, it can be found that ACF still achieves comparable performances compared with PureT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' To solve these limitations, we prepare to extend the com- putation resources like the GPU servers to build a novel pure vision global-local Transformer where ACF prior is used to learn hierarchical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' And then using this model to extract visual features for solving more vision-language tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=', by building a homogeneous ACF-based vision- language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Conclusion We propose a novel global-local Transformer named as Ada-ClustFormer (ACF) that can adaptively cluster the in- put elements for carrying self-attention (Self-ATT) to learn global-local contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Specifically, this is achieved by in- serting a clustering matrix into the Self-ATT layer, where the probability terms are calculated from the input data and thus ACF can adaptively cluster the elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Moreover, we use ACF to build an image captioning model to transfer more structural commonalities for better captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' The ex- periment results confirm the effectiveness of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' References [1] Abhaya Agarwal and Alon Lavie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Meteor: An automatic metric for mt evaluation with high levels of correlation with human judgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Proceedings of WMT-08, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [2] Mahtab Ahmed, Muhammad Rifayat Samee, and Robert E Mercer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' You only need attention to traverse trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Pro- ceedings of the 57th Annual Meeting of the Association for Computational Linguistics, pages 316–322, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [3] Peter Anderson, Basura Fernando, Mark Johnson, and Stephen Gould.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Spice: Semantic propositional image cap- tion evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In European conference on computer vision, pages 382–398.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Springer, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [4] Peter Anderson, Xiaodong He, Chris Buehler, Damien Teney, Mark Johnson, Stephen Gould, and Lei Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Bottom-up and top-down attention for image captioning and visual question answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Proceedings of the IEEE con- ference on computer vision and pattern recognition, pages 6077–6086, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [5] Peter W Battaglia, Jessica B Hamrick, Victor Bapst, Al- varo Sanchez-Gonzalez, Vinicius Zambaldi, Mateusz Ma- linowski, Andrea Tacchetti, David Raposo, Adam Santoro, Ryan Faulkner, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Relational inductive biases, deep learn- ing, and graph networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' arXiv preprint arXiv:1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='01261, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [6] Iz Beltagy, Matthew E Peters, and Arman Cohan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Long- former: The long-document transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' arXiv preprint arXiv:2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='05150, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [7] Boyu Chen, Peixia Li, Chuming Li, Baopu Li, Lei Bai, Chen Lin, Ming Sun, Junjie Yan, and Wanli Ouyang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Glit: Neural architecture search for global and local image transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 12–21, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [8] Marcella Cornia, Matteo Stefanini, Lorenzo Baraldi, and Rita Cucchiara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Meshed-memory transformer for image cap- tioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 10578– 10587, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [9] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Syl- vain Gelly, Jakob Uszkoreit, and Neil Houlsby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' An image is worth 16x16 words: Transformers for image recognition at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' ICLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [10] Zhiyuan Fang, Jianfeng Wang, Xiaowei Hu, Lin Liang, Zhe Gan, Lijuan Wang, Yezhou Yang, and Zicheng Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Injecting semantic concepts into end-to-end image captioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Pro- ceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 18009–18019, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [11] Longteng Guo, Jing Liu, Xinxin Zhu, Peng Yao, Shichen Lu, and Hanqing Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Normalized and geometry-aware self- attention network for image captioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 10327–10336, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [12] Simao Herdade, Armin Kappeler, Kofi Boakye, and Joao Soares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Image captioning: Transforming objects into words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, pages 11137–11147, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [13] Xiaowei Hu, Zhe Gan, Jianfeng Wang, Zhengyuan Yang, Zicheng Liu, Yumao Lu, and Lijuan Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Scaling up vision-language pre-training for image captioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Pro- ceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 17980–17989, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [14] Lun Huang, Wenmin Wang, Jie Chen, and Xiao-Yong Wei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Attention on attention for image captioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Proceedings of the IEEE International Conference on Computer Vision, pages 4634–4643, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [15] Huaizu Jiang, Ishan Misra, Marcus Rohrbach, Erik Learned- Miller, and Xinlei Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In defense of grid features for visual question answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Con- ference on Computer Vision and Pattern Recognition, pages 10267–10276, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [16] Andrej Karpathy and Li Fei-Fei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Deep visual-semantic align- ments for generating image descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recog- nition, pages 3128–3137, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [17] Ranjay Krishna, Yuke Zhu, Oliver Groth, Justin Johnson, Kenji Hata, Joshua Kravitz, Stephanie Chen, Yannis Kalan- tidis, Li-Jia Li, David A Shamma, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Visual genome: Connecting language and vision using crowdsourced dense image annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' International Journal of Computer Vi- sion, 123(1):32–73, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [18] Hwanhee Lee, Seunghyun Yoon, Franck Dernoncourt, Doo Soon Kim, Trung Bui, and Kyomin Jung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Vilbertscore: Evaluating image caption using vision-and-language bert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Proceedings of the First Workshop on Evaluation and Com- parison of NLP Systems, pages 34–39, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [19] Guang Li, Linchao Zhu, Ping Liu, and Yi Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Entan- gled transformer for image captioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), October 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [20] Jinpeng Li, Yichao Yan, Shengcai Liao, Xiaokang Yang, and Ling Shao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Local-to-global self-attention in vision trans- formers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' arXiv preprint arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='04735, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [21] Xiujun Li, Xi Yin, Chunyuan Li, Pengchuan Zhang, Xiaowei Hu, Lei Zhang, Lijuan Wang, Houdong Hu, Li Dong, Furu Wei, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Oscar: Object-semantics aligned pre-training for vision-language tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In European Conference on Computer Vision, pages 121–137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Springer, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [22] Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Doll´ar, and C Lawrence Zitnick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Microsoft coco: Common objects in context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In European conference on computer vision, pages 740–755.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Springer, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [23] Wei Liu, Sihan Chen, Longteng Guo, Xinxin Zhu, and Jing Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Cptr: Full transformer network for image captioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' arXiv preprint arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='10804, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [24] Ze Liu, Yutong Lin, Yue Cao, Han Hu, Yixuan Wei, Zheng Zhang, Stephen Lin, and Baining Guo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Swin transformer: Hierarchical vision transformer using shifted windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 10012–10022, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [25] Yunpeng Luo, Jiayi Ji, Xiaoshuai Sun, Liujuan Cao, Yongjian Wu, Feiyue Huang, Chia-Wen Lin, and Rongrong Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Dual-level collaborative transformer for image caption- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pages 2286–2293, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [26] Minh-Thang Luong, Hieu Pham, and Christopher D Man- ning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Effective approaches to attention-based neural machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' arXiv preprint arXiv:1508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='04025, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [27] Ron Mokady, Amir Hertz, and Amit H Bermano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Clip- cap: Clip prefix for image captioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' arXiv preprint arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='09734, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [28] Van-Quang Nguyen, Masanori Suganuma, and Takayuki Okatani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Grit: Faster and better image captioning transformer using dual visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' arXiv preprint arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='09666, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [29] Xuan-Phi Nguyen, Shafiq Joty, Steven Hoi, and Richard Socher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Tree-structured attention with hierarchical accumu- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In International Conference on Learning Representa- tions, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [30] Vicente Ordonez, Girish Kulkarni, and Tamara Berg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Im2text: Describing images using 1 million captioned pho- tographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Advances in neural information processing sys- tems, 24, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [31] Yingwei Pan, Ting Yao, Yehao Li, and Tao Mei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' X-linear attention networks for image captioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In CVPR, pages 10971–10980, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [32] Kishore Papineni, Salim Roukos, Todd Ward, and Wei-Jing Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Bleu: a method for automatic evaluation of machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Proceedings of the 40th annual meeting of the Association for Computational Linguistics, pages 311–318, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [33] Samrudhdhi B Rangrej, Kevin J Liang, Tal Hassner, and James J Clark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Glitr: Glimpse transformers with spatiotem- poral consistency for online action prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' arXiv preprint arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='13605, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [34] S Ren, K He, R Girshick, and J Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Towards real-time ob- ject detection with region proposal networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Advances in neural information processing systems, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [35] Steven J Rennie, Etienne Marcheret, Youssef Mroueh, Jerret Ross, and Vaibhava Goel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Self-critical sequence training for image captioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 7008–7024, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [36] Lin CY ROUGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' A package for automatic evaluation of summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Proceedings of Workshop on Text Summa- rization of ACL, Spain, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [37] Piyush Sharma, Nan Ding, Sebastian Goodman, and Radu Soricut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Conceptual captions: A cleaned, hypernymed, im- age alt-text dataset for automatic image captioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Pro- ceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 2556–2565, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [38] Ying Hua Tan and Chee Seng Chan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Phrase-based image caption generator with hierarchical lstm network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Neurocom- puting, 333:86–100, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [39] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszko- reit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Attention is all you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Advances in neural information processing systems, 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [40] Ramakrishna Vedantam, C Lawrence Zitnick, and Devi Parikh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Cider: Consensus-based image description evalua- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4566–4575, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [41] Oriol Vinyals, Alexander Toshev, Samy Bengio, and Du- mitru Erhan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Show and tell: A neural image caption gen- erator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3156–3164, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [42] Yiyu Wang, Jungang Xu, and Yingfei Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' End-to-end trans- former based model for image captioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Proceedings of the AAAI Conference on Artificial Intelligence, pages 2585– 2594, Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [43] Yau-Shian Wang, Hung-Yi Lee, and Yun-Nung Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Tree transformer: Integrating tree structures into self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' arXiv preprint arXiv:1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='06639, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [44] Chuhan Wu, Fangzhao Wu, Tao Qi, and Yongfeng Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Hi-transformer: hierarchical interactive transformer for effi- cient and effective long document modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' arXiv preprint arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content='01040, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [45] Tiantao Xian, Zhixin Li, Canlong Zhang, and Huifang Ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Dual global enhanced transformer for image captioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Neural Networks, 148:129–141, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [46] Kelvin Xu, Jimmy Ba, Ryan Kiros, Kyunghyun Cho, Aaron Courville, Ruslan Salakhudinov, Rich Zemel, and Yoshua Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Show, attend and tell: Neural image caption gen- eration with visual attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In International conference on machine learning, pages 2048–2057.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' PMLR, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [47] Jianwei Yang, Chunyuan Li, Pengchuan Zhang, Xiyang Dai, Bin Xiao, Lu Yuan, and Jianfeng Gao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Focal attention for long-range interactions in vision transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 34:30008–30022, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [48] Xu Yang, Chongyang Gao, Hanwang Zhang, and Jianfei Cai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Auto-parsing network for image captioning and visual ques- tion answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Interna- tional Conference on Computer Vision, pages 2197–2207, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [49] Xu Yang, Kaihua Tang, Hanwang Zhang, and Jianfei Cai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Auto-encoding scene graphs for image captioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Pro- ceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 10685–10694, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [50] Xu Yang, Hanwang Zhang, Guojun Qi, and Jianfei Cai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Causal attention for vision-language tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pat- tern Recognition, pages 9847–9857, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [51] Ting Yao, Yingwei Pan, Yehao Li, and Tao Mei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Hierar- chy parsing for image captioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 2621–2629, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [52] Pengchuan Zhang, Xiujun Li, Xiaowei Hu, Jianwei Yang, Lei Zhang, Lijuan Wang, Yejin Choi, and Jianfeng Gao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Vinvl: Revisiting visual representations in vision-language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 5579– 5588, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [53] Hengshuang Zhao, Jiaya Jia, and Vladlen Koltun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Explor- ing self-attention for image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 10076–10085, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' [54] Luowei Zhou, Hamid Palangi, Lei Zhang, Houdong Hu, Ja- son Corso, and Jianfeng Gao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' Unified vision-language pre- training for image captioning and vqa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} +page_content=' In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pages 13041–13049, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQf__8r/content/2301.01955v1.pdf'} diff --git a/BNAzT4oBgHgl3EQfhv2_/vector_store/index.faiss b/BNAzT4oBgHgl3EQfhv2_/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..22a2c7f49cd44acc20aac028ee0f0a49b1edf247 --- /dev/null +++ b/BNAzT4oBgHgl3EQfhv2_/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:479ebe0d6d8095def64011e9594369647222906387f5a53593dbd9c410cedfe9 +size 1835053 diff --git a/BNAzT4oBgHgl3EQfhv2_/vector_store/index.pkl b/BNAzT4oBgHgl3EQfhv2_/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..9eb861f5e9ae5c80d1e28466a785aa9113f97be1 --- /dev/null +++ b/BNAzT4oBgHgl3EQfhv2_/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:bdfd482e3047cbf9b559ffd66d0bcfd4b46dc4400208af4f7cf794f9acedf8aa +size 86954 diff --git a/BdE5T4oBgHgl3EQfTA_5/content/tmp_files/2301.05534v1.pdf.txt b/BdE5T4oBgHgl3EQfTA_5/content/tmp_files/2301.05534v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9e8626a06f1cdedff911ad5634b48b4aacadbbe9 --- /dev/null +++ b/BdE5T4oBgHgl3EQfTA_5/content/tmp_files/2301.05534v1.pdf.txt @@ -0,0 +1,2442 @@ +arXiv:2301.05534v1 [math.DS] 13 Jan 2023 +GLOBAL DYNAMICS AND PERSPECTIVES ON SINGULARITIES OF +HOLOMORPHIC FOLIATIONS +JULIO REBELO AND HELENA REIS +Abstract. In dimensions greater than or equal to 3, the local structure of a singular holo- +morphic foliation conceals a globally defined foliation on the projective space of dimension one +less. In this paper, we will investigate how the global dynamics of the latter foliation exerts +influence on several problems that apparently have a purely local nature. In the course of the +discussion, a few recent results and open problems in the area will be reviewed as well. +Contents +1. +Introduction +1 +2. +Basics in the local theory of foliations and the special case of dimension 2 +3 +3. +Splitting the problem: core dynamics and resolution +7 +4. +Resolution theorems in dimension 3 +14 +5. +Invariant analytic sets +21 +6. +Semicomplete vector fields, automorphism groups, and separatrices +30 +References +42 +1. Introduction +All foliations considered in this work are holomorphic and (possibly) singular. Whereas our +main object are 1-dimensional holomorphic foliations and holomorphic/meromorphic vector +fields, foliations of codimension 1 will also play a role in the discussion especially when the am- +bient is of dimension 3, see Section 2 for accurate definitions. Foliations of dimension 1 defined +on some complex manifold M will typically be denoted by F while D will stand for codimen- +sion 1 foliations. The purpose of this paper is to discuss recent results and open problems in the +local theory of 1-dimensional foliations when the ambient manifold M has dimension 3 though, +occasionally, results and questions in higher dimensions will also be included. +Foliations defined on complex surfaces, i.e. complex manifolds of dimension 2, are basically +left aside in this paper largely due to the fact that their local theory is much more advanced +than their higher dimensional counterparts. Indeed, these singularities are only mentioned in +Section 2 and, yet, with the simple purpose of identifying a few issues that make them so +special and amenable to very detailed analysis. In doing so, we will be able to single out one +of the most fundamental issues guiding the discussion conducted here: the presence of a global +dynamical phenomenon intrinsically attached to germs F of 1-dimensional foliations on Cn +provided that n ≥ 3. In slightly vague though more incisive words, the understanding of a germ +of 1-dimensional foliation F on Cn, n ≥ 3, passes through the description of a foliation defined +on CPn−1 which, as a global object, may exhibit a wild dynamical behavior (cf. Section 3). +The global foliation in question will usually be referred to as the core foliation of the (local) +2010 Mathematics Subject Classification. Primary 34M35, 32M25; Secondary 34M45, 32M05. +1 + +2 +J. REBELO AND H. REIS +foliation F. We will also use the phrase core dynamics of F to refer to the dynamics of the +core foliation associated with F. +The common thread of this paper is the existence and implications of a global dynamical +system inherently attached to the structure of a singularity of a 1-dimensional, holomorphic +foliation defined on (Cn, 0) provided that n ≥ 3. Basically, we will discuss which types of results +can be proved if the above mentioned dynamics can accurately be described and, similarly, which +general problems may provide us with the tools to ensure this dynamics is “tame enough” to +be described, while bearing in mind that in full generality this dynamics can be extremely wild. +The paper is structured as follows. In Section 2, we introduce standard terminology and recall +some basic features of singular foliations, in particular pointing out fundamental issues setting +apart foliations of dimension 1 and foliations of codimension 1. Then we focus on the special +case of singularities of foliations defined on (C2, 0). Whereas this case is clearly distinguished by +the fact that the foliations are simultaneously of dimension 1 and of codimension 1, we discuss +to a rather non-trivial extent the main 2-dimensional issues allowing for the existence of such +a sophisticated theory covering truly fine issues. +In Section 3, we introduce a fundamental object that exists for singularities of (1-dimensional) +foliations defined on (Cn, 0) provided that n ≥ 3, namely: the core dynamics. This is a global +foliation defined on CPn−1 whose (global) dynamics poses a fundamental obstacle towards the +full understanding of the initial singular point. In particular, we explain how this core dynamics +plays a major role in problems about existence of separatrices for codimension 1 foliations on +(C3, 0). Also, we show how its very existence basically rules out any hope of achieving a full +understanding of large classes of singular points. +The remainder of this survey is devoted to more advanced material, in particular touching on +quite a few open problems of current interest. Section 4 contains a detailed review of resolution +theorems for singular points of 1-dimensional foliations on (C3, 0). The first definitive resolution +theorem in this context was obtained by McQuillan and Panazzolo in [37] which, in turn, relies +heavily on Panazzolo’s algorithm introduced in [39]. More recently, a different proof based on +Zariski general point of view was obtained in [49] which can be seen as the completion of the +previous work carried out by Cano-Roche-Spivakovsky [9]. Despite the undisputed importance +of resolution theorem, it seems these results are still not as widely known as one would expect +and, for this reason, we thought useful to conduct a thorough discussion about the content of +the resolution theorems in [37] and in [49], highlighting virtues and potential limitations. +In Section 5, we consider the fundamental problem of existence of separatrices that, roughly +speaking, concerns the existence of germs of analytic sets invariant by (germs of) singular +foliations. The discussion is essentially conducted in (C3, 0). Expanding on the discussion of +Section 3, we consider the existence of separatrices for codimension 1 foliation spanned by two +commuting vector fields and state Theorem 5.3 affirmatively answering this question. We also +detail the general strategy for proving this theorem which, in turn, emphasizes a few often +overlooked points in resolution theorems for foliations as well as the major role played by the +general question of “taming a core dynamics”. The second part of this section, we review some +results on the existence of separatrices for foliations of dimension 1. The nature of this second +problem is far more topological/geometric and “core dynamics” plays a much smaller role. Yet, +some of the results will find applications in the last section. +Finally, in Section 6 we discuss a particular class of singularities of foliations of dimension 1, +namely those supporting semicomplete vector fields. +Albeit small in an appropriate sense, +this class of singularities has rather distinguished properties and quite a few applications that +make it worth studying. The section will precisely begin with proper definitions and a general +discussion of applications. +Once the basic setting is in place, we will go on to discuss two + +GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS +3 +fundamental problems on the area: the first problem can vaguely be stated by asking how +wild can the core dynamics be in this class of foliations? The main results here stem from the +seminal paper of A. Guillot about Halphen vector fields and their role in SL (2, C)-actions, see +[17]. The second problem aims at quantifying how “degenerate” a singularity in this class can +be. This second problem stems from a well known question raised long ago by E. Ghys and the +topic has applications in the study of automorphism groups of compact complex manifolds. +Acknowledgment. We are grateful to many of our colleagues for several discussion over the +years. A special thanks is due to D. Panazzolo who has helped us to understand many subtle +points of his fundamental work on desingularization theorems. We also thank J.-F. Mattei for +many discussions and explanations about the vast and fundamental work in singularity theory +he has accomplished with his collaborators. +J. Rebelo and H. Reis were partially supported by CIMI through the project “Complex +dynamics of group actions, Halphen and Painlev´e systems”. H. Reis was also partially supported +under the project “Means and Extremes in Dynamical Systems” with reference PTDC/MAT- +PUR/4048/2021 and also by CMUP, member of LASI, which is financed by national funds +through FCT Funda¸c˜ao para a Ciˆencia e Tecnologia, I.P., under the project UIDB/00144/2020 +as also supported . +2. Basics in the local theory of foliations and the special case of dimension 2 +It is convenient to begin by recalling the definition of 1-dimensional singular holomorphic +foliation. First, let X = f1∂/∂x1 + · · · + fn∂/∂xn be a non-trivial holomorphic vector field +defined on an open set V of Cn. The singular set Sing (X) of X is then given by �n +i=1{fi = 0}. +It is a (proper) analytic subset of V and it is well known that Sing (X) has codimension 1 if +and only if the coordinate functions fi admit a non-trivial common factor. +We are then able to define singular holomorphic foliations as they will be considered through- +out this work. +Let M be a complex manifold and consider a covering {(Uk, ϕk)} of M by +coordinate charts. We denote by n the dimension of M so that ϕk(Uk) is an open set of Cn. +Definition 2.1. Let M and {(Uk, ϕk)} be as above. A singular holomorphic foliation F of +dimension 1 on M consists of a collection of holomorphic vector fields Yk satisfying the following +conditions: +• For every k, Yk is a holomorphic vector field defined on ϕk(Uk) ⊂ Cn whose singular set +has codimension at least 2. +• Whenever Uk1∩Uk2 ̸= ∅, we have Yk1 = gk1k2.(ϕk2◦ϕ−1 +k1 )∗Yk2 for some nowhere vanishing +holomorphic function gk1k2. +The singular set Sing (F) of a foliation F is then defined as the union over k of the sets +ϕ−1 +k (Sing (Yk)) ⊂ M. Therefore the singular set of any holomorphic foliation has codimension +at least two. In particular, a foliation has no divisor of zeros. +Conversely, we say that a holomorphic vector field Y is a local representative of the 1- +dimensional foliation F if Y is tangent to F and the singular set of Y has codimension at +least 2. It is clear that representative vector fields are locally unique up to multiplication by +an invertible holomorphic function. +Analogously, we might also consider a differential 1-form ω on V ⊆ Cn, ω = g1dx1 + · · · + +gndxn. Again the singular set Sing (ω) of ω is given by the intersection �n +i=1{gi = 0} and it is +an analytic set of V which has codimension 1 if and only if there is a non-trivial common factor +for the functions g1, . . . , gn. Away from its singular points, the kernel of ω defines a distribution +of complex hyperplanes on V . If in Definition 2.1 we replace “local vector fields” by “integrable +local 1-forms”, we obtain the notion codimension 1 foliations. More precisely, we have: + +4 +J. REBELO AND H. REIS +Definition 2.2. Let M be a complex manifold and {(Uk, ϕk)} a covering of M by coordinate +charts. A singular holomorphic foliation D of codimension 1 on M consists of a collection of +differential 1-forms Ωk satisfying the following conditions: +• For every k, Ωk is a differential 1-form defined on ϕk(Uk) ⊂ Cn with singular set of +codimension at least 2 and such that Ωk ∧ dΩk vanishes identically. +• Whenever Uk1∩Uk2 ̸= ∅, we have Ωk1 = gk1k2.(ϕk2◦ϕ−1 +k1 )∗Ωk2 for some nowhere vanishing +holomorphic function gk1k2. +In particular the singular set Sing (D) of D again has codimension at least 2. The notion of +representative 1-form for a codimension 1 foliation D is defined analogously to the notion of +representative vector fields in the case of foliations with dimension 1. +Whereas our main focus here is on germs of foliations, or in slightly more concrete terms, +on foliations defined on a neighborhood of the origin in Cn, the reader will notice that the +global point of view considered in Definitions 2.1 and 2.2 is really indispensable to investigate +the local structure of the singular point. Indeed, globally defined foliations - and in particular +the “global dynamical phenomenon” mentioned in the Introduction - will come to fore in the +context of birational theory of foliations which is needed, for example, if one seeks to “resolve +singular points”. +It is also convenient to complement the above definitions with a couple of comments. +Remark 2.3. Already on (Cn, 0), n ≥ 3, it is easy to see a first fundamental difference between +foliations of dimension 1 and foliations of codimension 1 arising from the Frobenius condition. +To formulate this, note that any choice of local holomorphic functions f1, . . . , fn naturally gives +rise to two (singular) distributions: one of lines and one of hyperplanes. In fact, to the collection +of functions f1, . . . , fn, we may associate the vector field Y = f1∂/∂x1 + · · · + fn∂/∂xn or the +1-form Ω = f1dx1 + · · · + fndxn. Whereas the local integral curves of Y always yield a 1- +dimensional foliation F, the Frobenius condition for Ω to yield a codimension 1 foliation is +non-trivial and amounts to requiring the 3-form Ω ∧ dΩ to vanish identically which, in turn, +leads to a highly non-trivial PDE system involving the functions f1, . . . , fn. +Remark 2.4. In general, foliations of dimension 1 are very abundant, at least in algebraic +manifolds, and they may have an extremely complicated dynamical behavior, more on this in +Section 3, see also [24], [28]. This contrasts with the case of codimension 1 foliations that are +far more rigid and in several cases amenable to classification, at least at conjectural level, all +codimension 1 foliations on, say, CPn should be transversely homogeneous or can be obtained +as a suitable pull-back of a foliaion defined on a surface. For an interesting discussion of several +of global aspects of codimension 1 foliations, we refer the reader to [58]. +A basic object in the local theory of foliations that has largely motivated its early development +is the notion of separatrix. Although the definition of separatrix depends on the dimension of +the foliation, the cases of foliations of dimension 1 and of codimension 1 can naturally be +formulated together. +Definition 2.5. Let F (resp. D) be a foliation of dimension 1 (resp. codimension 1) on (Cn, 0). +A separatrix S for F (resp. D) is the germ of an irreducible analytic set of dimension 1 (resp. +codimension 1) containing 0 ∈ Cn and invariant by F (resp. D). +Separatrices are objects of natural interest since they fit the framework of “invariant man- +ifolds” in dynamical systems. In particular, their presence may enable one to reduce the di- +mension of the phase space of the system in question. Yet, in the local theory of foliations as +discussed here, the notion of separatrix first appeared in the classical work of Briot and Bouquet +[2] where it was claimed that every foliation on (C2, 0) admits at least one separatrix. Much + +GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS +5 +later, R. Thom has sought to generalize the existence of separatrices for every codimension 1 +foliation on (C3, 0). The first example of a codimension 1 foliation on (C3, 0) without separatrix +was, however, found by J.-P. Jouanolou [26]. As it will be seen in the next section, Jouanolou’s +counterexample relies on the core dynamics of certain 1-dimensional foliations on (C3, 0). Let +us also point out that a gap in the arguments of Briot and Bouquet was later found and the +existence of separatrices for foliations on (C2, 0) was firmly established by Camacho and Sad in +[5]. +Another fundamental notion in the theory of singularities of foliations is the notion of eigen- +values of a foliation at a singular point. To abridge the discussion, for the time being let us +restrict ourselves to the case of foliations of dimension 1 (see Section 4 for a more general +discussion). Without loss of generality, it suffices to consider the case of foliations F of dimen- +sion 1 defined on (Cn, 0). Given F as above, up to reducing the neighborhood of the origin, +there is a holomorphic vector field Y whose zero-set has codimension at least 2 and such that +F is nothing but the foliation induced by the local orbits of Y . As mentioned, Y is said to be +a representative of F and, while Y is not unique, two representative vector fields for the same +foliation F differ by multiplication by an invertible holomorphic function. +Definition 2.6. Let F be a 1-dimensional holomorphic foliation on (Cn, 0) and let Y denote a +representative vector field for F. Assume that F is singular at the origin, i.e., Y (0) = 0. Then +the eigenvalues of F at 0 ∈ Cn are the eigenvalues of the Jacobian matrix D0Y . +Since Y is well defined up to multiplication by an invertible holomorphic function, there fol- +lows that the eigenvalues of F at 0 ∈ Cn are well defined only up to simultaneous multiplication +by a non-zero constant. +2.1. Blow-ups and dicritical singularities. Blow-ups are a standard tool to produce non- +trivial birational maps and to understand the local structures of singular points, whether these +are “singularities of the ambient space” or “singularities of a foliation on a smooth space”. +The transform of a foliation under a blow-up map is called the blow-up of the foliation. The +blown-up space, however, contains an exceptional divisor which may or may not be invariant by +the transformed foliation. This issue gives rise to the notion of dicritical foliation at a singular +point. +Definition 2.7. Let M be a complex manifold equipped with a holomorphic foliation H and +consider a blow-up map π : � +M → M centered at C ⊂ Sing(H), where Sing(H) stands for the +singular set of H. The foliation H is said to be dicritical with respect to π if its corresponding +blow-up � +H does not leave the exceptional divisor π−1(C) invariant. +Whenever no misunderstanding is possible, we will simply say that a given foliation is, or +is not, dicritical without specifically mentioning to the blow-up map. Also, for most of our +discussion, it will suffice to consider blow-ups centered at single points (sometimes called one- +point blow-ups). +For this type of blow-up, the characterization of 1-dimensional dicritical +foliations is very simple. More precisely, let F be a 1-dimensional foliation on (Cn, 0) and fix a +representative vector field Y of F. Denote by Yk the non-zero homogeneous component of least +degree in the Taylor series of Y based at 0 ∈ Cn. Then, we have: +Lemma 2.8. The foliation F is dicritical with respect to the blow-up centered at 0 ∈ Cn if and +only if Yk is a multiple of the radial vector field R = x1∂/∂x1 + · · · + xn∂/∂xn. +Proof. It suffices to compute the pull-back of Y in the coordinates (x1, u2, . . . , un) for the blow- +up of Cn where the blow-up map π is given by π(x1, u2, . . . , un) = (x1, x1u2, . . . , x1un), cf. for +example [23] or [43]. +□ + +6 +J. REBELO AND H. REIS +In more general terms, a foliation F is said to be dicritical at a center C if there exists a +sequence of blow-ups beginning at C and leading to a foliation which does not leave all the +irreducible components of the global exceptional divisor invariant (for details see Section 4). +Since a blow-up map is proper, and therefore so is a composition of blow-up maps, there follows +from Remmert’s theorem that a leaf of the foliation �F transverse to (a component of) the +exceptional divisor must project to a separatrix for the initial foliation F. Thus we obtain the +following simple characteristic of 1-dimensional dicritical foliations: +Lemma 2.9. If the foliation F at the center C is dicritical, then the union of separatrices of +F through points of C yields a set with non-empty interior. +□ +The converse to Lemma 2.9 is known to hold for ambient spaces of dimension up to 3, and +it is a simple consequence of “resolution theorems”, cf. Section 4. Whereas the result is likely +to hold in general, a proof of this statement dispensing with “resolution” results seems to be +still lacking in the literature. +The above lemmas show that 1-dimensional dicritical foliations are, somehow, very special. +In particular, a “generic foliation” is not dicritical at their singular points. +Also, owing to +Lemma 2.9, for most of the problems discussed here involving 1-dimensional foliations, we can +assume without loss of generality that the foliation in question is not dicritical. +Remark 2.10. Examples of dicritical foliations are far more abundant when we consider codi- +mension 1 foliation in ambient spaces of dimension 3,. In particular, it is unclear if there is any +reasonable sense in claiming that a “generic foliation” is not dicritical. In fact, a good source +of examples of dicritical foliations consists of exploiting the affine Lie algebra generated by a +homogeneous polynomial vector field and by the radial vector field, see Section 3. +2.2. Singularities of foliations on (C2, 0). As mentioned, singularities of foliations on (C2, 0) +are the object of a highly developed theory, at least in the very general setting of non-dicritical +foliations. In this paragraph, we shall collect some reasons that allowed so much progress in +this topic and compare them with the situation of foliations on (C3, 0). +(A) Seidenberg theorem. It is commonly accepted that no general theorem in singularity +theory can be proved without relying on a suitable “desingularization theorem”. In the theory +of foliations, however, it is not possible in general to actually desingularize a foliation, i.e., to +obtain a non-singular model of the foliation up to birational transformations. In fact, whereas +the phrase desingularization theorem is sometimes used as an abuse of language, a more accurate +terminology would be reduction of singularities theorem. In other words, rather than looking +for a non-singular foliation, we look for a foliation whose singular points are as “well behaved +as possible”. Typically, we will look for a foliation all of whose singular points are elementary, +i.e., all of them have at least one eigenvalue different from zero. +Seidenberg theorem [55] provides a suitable procedure to reduce the singularities of holo- +morphic foliations on (C2, 0). +Let F denote a singular holomorphic foliation defined on a +neighborhood of (0, 0) ∈ C2. Seidenberg theorem asserts the existence of a finite sequence of +blow-up maps, along with transformed foliations Fi (i = 1, . . . , n) +F = F0 +π1 +←− F1 +π2 +←− · · · +πn +←− Fn +such that the following holds: +• Each blow-up map πi (i = 1, . . . , n) is centered at a singular point of Fi−1. +• All singular points of Fn are elementary, i.e., the foliation Fn possesses at least one +eigenvalue different from zero at each of them. + +GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS +7 +Denote by D1, . . . , Dn the irreducible components of the total exceptional divisor associated +with Fn. Each Di is therefore a rational curve with strictly negative self-intersection and the +corresponding Dynkin diagram is a tree. +(B) A global pseudogroup - Mattei-Moussu technique. Assume next that the foliation +F is not dicritical. Then, for each i = 1, . . . , n, Di \ Sing (Fn) is a regular leaf of Fn, where +Sing (Fn) stands for the singular set of Fn. In particular, all non-trivial dynamics associated +with the foliation Fn is of transverse nature. +Moreover this transverse dynamics naturally +arises from the holonomy representations of each of the leaves Di \ Sing (Fn), i = 1, . . . , n. In +turn, at least to a considerable extent, the dynamics of these representations can be merged +together through the argument of “passage of corners” (a.k.a. “Dulac transform”), whenever +Di ∩ Dj ̸= ∅. +The preceding can be summarized by saying that all singular points of Fn are “dynamically +connected” in the sense that their local dynamics blend together in a nice pseudogroup of maps +of (C, 0). Furthermore, the dynamics of this pseudogroup encodes virtually all the information +on the local structure of the initial foliation F. +The method described above to investigate the singularities of foliations on (C2, 0) was very +much set up in the seminal paper by Mattei and Moussu [35]. This technique has proven time +and again to be extremely effective in a variety of situations in dimension 2, see [5] and [19] for +two examples of problems whose solutions have involved this type of setup. In the next section +we will discuss how far this approach can be generalized to higher dimensions. +(C) Dynamics of pseudogroups acting on (C, 0). Although for many problems this issue +plays a relatively minor role, let us still point out that the dynamics of pseudogroups acting on +(C, 0) is itself a highly developed topic. This type of dynamics was first investigate by Huddai- +Verenov [22] and then by Il’yashenko in [24] where a “generic situation” of groups generated +by hyperbolic diffeomorphisms was considered. In contrast, in [35], the authors have dealt with +subgroups all of whose orbits are finite. An absolute breakthrough then came with the works of +Shcherbakov and of Nakai about general non-solvable subgroups, see [53], [54], [38]. The reader +may consult [42] and references therein for a more complete account of these dynamics in the +non-solvable case whereas solvable pseudogroups are discussed in detail in [11]. +Remark 2.11. It should be pointed out that much progress in terms of construction of moduli +spaces for foliations on (C2, 0) and in describing the topology of leaves has been made in recent +years, chiefly by Mar´ın, Mattei, and Salem. While these aspects will not be discussed in this +survey which is mostly devoted to higher dimensional situations. Yet, the reader interested in +the topology of leaves will find more up-to-date information in [30], [31], and [57]. As to the +construction of moduli spaces, we refer to [32] and to the preprints [33] and [34]. +3. Splitting the problem: core dynamics and resolution +The main object of this section are 1-dimensional foliations defined around the origin of Cn, +for n ≥ 3. Yet, most of the discussion can be conducted without loss of generality in the case +n = 3. +It is useful to begin by recalling some well known facts about foliations on complex projective +spaces. Let CPn be viewed as the space of radial lines through the origin of Cn+1 and denote by +Π : Cn+1 \ {0} → CPn the canonical projection. Also, for λ ∈ C∗, denote by hλ : Cn+1 → Cn+1 +the homothety defined by hλ(x1, . . . , xn+1) = (λx1, . . . , λxn+1). Finally let R denote the radial +vector field R = x1∂/∂x1 + · · · + xn+1∂/∂xn+1 and consider a homogeneous polynomial vector + +8 +J. REBELO AND H. REIS +field +X = P1 +∂ +∂x1 ++ · · · + Pn+1 +∂ +∂xn+1 +of degree d on Cn+1. In other words, each Pi is a degree d homogeneous polynomial, for every +i = 1, . . . , n + 1. In what follows X is always assumed to satisfy the following conditions: +(1) The singular set of X on Cn+1 has codimension at least 2. +(2) The vector fields X and R are linearly independent at generic points. +Next note that we have +h∗ +λX = λd−1X +so that the vector fields h∗ +λX and X are everywhere parallel for any fixed value of λ ∈ C∗. In +particular, if p ∈ Cn+1 is a point at which X(p) and R(p) are linearly independent, then X(p) +induces a direction in Tq=Π(p)CPn which is well defined in the sense that it does not depend on +p ∈ Π−1(q). From this, it easily follows that X induces a singular holomorphic foliation F on +CPn in the sense of Definition 2.1. A standard application of Serre’s GAGA principle yields +a type of converse for the above construction, namely the following proposition holds, cf. for +example [23], [43]. +Proposition 3.1. Let F denote a singular holomorphic foliation on CPn. Then, there exists a +homogeneous polynomial vector field X on Cn+1 having singular set of codimension at least 2 +and inducing the foliation F on CPn by means of the above described construction. +□ +Whereas, given F, the mentioned homogeneous vector field X of Proposition 3.1 is not +uniquely defined, two homogeneous polynomial vector fields having singular set of codimension +at least 2 and inducing the same foliation on CPn must have the same degree. Thus we can +talk about the degree of a foliation on CPn as follows: +Definition 3.2. The degree of a foliation F on CPn is the degree of a homogeneous polynomial +vector field on Cn+1 having singular set of codimension at least 2 and inducing F in CPn viewed +as the space of radial lines of Cn+1. +Naturally blow-ups provide an alternative way to realize the foliation induced on CPn by +a homogeneous polynomial vector field X on Cn+1. Let �Cn+1 stand for the blow-up of Cn+1 +at the origin and consider a homogeneous vector field X as above on Cn+1. The blow up � +X +of X induces on �Cn+1 the blow up �F of the the foliation F induced by X on Cn. Since, by +assumption, X is not everywhere parallel to the radial vector field R, there follows that the +foliation �F leaves invariant the exceptional divisor π−1(0) ≃ CPn. The restriction of �F to the +exceptional divisor π−1(0) ≃ CPn can then naturally be identified with the foliation induced by +X on CPn - viewed as the space of radial lines of Cn+1 - by means of the preceding construction. +Note that the blow up construction does not really requires the vector field to be homoge- +neous. In fact, as in Lemma 2.8, consider a holomorphic vector field Y defined around the +origin of Cn+1 whose Taylor series takes on the form Y = �∞ +i=k Yi, where Yi stands for the +homogeneous component of degree i of this Taylor series and Yk is not identically zero. As in +Lemma 2.8, we assume that Yk is not everywhere parallel to the radial vector field R. The +blow up of Y induces a holomorphic foliation �F on a neighborhood of the exceptional divisor +π−1(0) ⊂ �Cn+1. Moreover, since Yk is not a multiple of R, this foliation leaves π−1(0) ≃ CPn +invariant and, in addition, it is immediate to check that the restriction of �F to π−1(0) coincides +with the restriction to π−1(0) of the foliation induced on �Cn+1 by the blow up of Yk (alone). +In particular, the restriction of �F to π−1(0) is identified with the foliation induced by the +homogeneous vector field Yk on CPn viewed as the space of lines of Cn+1. +The preceding motivates the following definition. + +GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS +9 +Definition 3.3. Let F be a 1-dimensional holomorphic foliation defined around the origin of +Cn and assume that the blow up �F of F at the origin leaves the exceptional divisor π−1(0) +invariant. Then the foliation induced on CPn−1 ≃ π−1(0) by the restriction of �F is called the +core foliation of F and its global dynamics is referred to as the core dynamics of F. +Again, if Y is a representative of F and Yk is as above (Y = �∞ +i=k Yi), the preceding +then shows that the core foliation of F is nothing but the foliation induced on CPn−1 by the +homogeneous vector field Yk. +3.1. 1-dimensional foliations and dicritical codimension 1 foliations on C3. The pre- +ceding discussion about foliations on projective spaces also applies to the case of codimension 1 +dicritical foliations on (Cn, 0). +To be more precise, codimension 1 dicritical foliation D on +(Cn, 0) also induces through the one-point blow-up centered at the origin a foliation on CPn−1, +that will also be called the core foliation of D. It is fair to say that this phenomenon and the +corresponding dynamics were first exploited by Jouanolou [26] in his famous counterexample +to a question posed by R. Thom. We shall review this issue below and go somewhat further by +exploiting the results in [28] to see how difficult the situation may become. +In the sequel, we set n = 3 to abridge notation. First, let us characterize codimension 1 +foliations that are dicritical for the blow-up of C3 centered at the origin. Since the lemma +below does not seem to be accurately stated in the literature, a detailed - albeit straightforward +- proof is included below. +Lemma 3.4. Assume that D is a singular codimension 1 foliation defined on (C3, 0) and denote +by �D its blow-up centered at the origin. Then the exceptional divisor π−1(0) ≃ CP2 is invariant +under �D if and only if no holomorphic vector field Z tangent to D admits a first non-zero +homogeneous component (at the origin) that is a multiple of the radial vector field R. +Proof. Let D be given by a holomorphic 1-form Ω = F dx + G dy + H dz whose singular set has +codimension at least 2. Denote by Ωk the first non-zero homogeneous component of Ω at the +origin, where k stands for the degree of Ωk. Next, let Ωk = F kdx + Gkdy + Hkdz. A direct +inspection shows that π−1(0) is not invariant by �D if and only if +(1) +xF k + yGk + zHk = 0 . +Now, note that if Z is any vector field tangent to D, and whose first non-zero homogeneous +component is denoted by Zl, then Zl naturally provides a solution for {Ωk = 0}, i.e., we have +Ωk.Zl = 0. However, if Zl happens to be a multiple of the radial vector field, then Ωk.Zl = 0 +is tantamount to Equation (1) which is thus satisfied. Hence, the exceptional divisor is not +invariant by �D. +To show that the existence of a vector field Z satisfying the above mentioned conditions is +also necessary, we proceed as follows. Assume that D is dicritical, i.e., that Equation (1) holds +and denote by ∧ the standard exterior power of two vectors on C3. Next define a vector field +v by letting v(p) = R(p) ∧ (F(p), G(p), H(p)) and then set Z(p) = v(p) ∧ (F(p), G(p), H(p)). +Clearly Z is tangent to the foliation D. To complete the proof of the lemma, it suffices to check +that the first non-zero homogeneous component of Z at the origin is a multiple of the radial +vector field. For this, note that we have +(2) Z = (zFH + yFG − x(H2 + G2) , xFG + zHG − y(F 2 + H2) , yHG + xFH − z(G2 + F 2)) +In particular the order of Z at the origin is at least 2k + 1. The homogeneous component of +degree 2k + 1 is, in turn, given in vector notation by +(xF k + yGk + zHk)(F k, Gk, Hk) − ((F k)2 + (Gk)2 + (Hk)2)(x, y, z) . + +10 +J. REBELO AND H. REIS +In view of Equation (1), we conclude that the component of degree 2d + 1 of Z at the origin is +given by −((F k)2 + (Gk)2 + (Hk)2)R. +To finish the proof of the lemma it suffices to show that the polynomial (F k)2+(Gk)2+(Hk)2 +cannot vanish identically. This, however, can easily be done by using the variables (x, t, u) where +the blow-up map becomes Π(x, t, u) = (x, xt, xu). In these variables, the dicritical condition +(i.e. Equation (1)) means that F k(1, t, u) + tGk(1, t, u) + uHk(1, t, u) must vanish identically. +Now, suppose for a contradiction that (F k)2 + (Gk)2 + (Hk)2 is also identically zero. Then the +two equations taken together imply that (t2 + 1)(Gk)2 + 2tu(Gk)(Hk) + (u2 + 1)(Hk)2 vanishes +identically as well. By solving the corresponding last equation for Gk, we derive a contradiction +with the fact that Gk is itself a polynomial in the variables t, u. The lemma is proved. +□ +Next, let us consider again a homogeneous polynomial vector field X on C3 satisfying condi- +tions (1) and (2) in the previous subsection, i.e. the singular set of X has codimension at least 2 +and the vector fields X and R are linearly independent at generic points (note that in the case +of homogeneous vector fields of degree at least 2, conditions (1) and (2) are equivalent). Since +X is homogeneous, we have +[R, X] = (d − 1)X +where d stands for the degree of X. Thus the pair X and R generates the Lie algebra of the +affine group. In particular the distribution of planes (of dimension 2) spanned by X and R is +involutive and hence integrable. Let us then denote by D the codimension 1 foliation spanned +by X and R. +Let �D stands for the blow-up of D centered at the origin so that �D is defined on �C3. Owing to +Lemma 3.4, the exceptional divisor π−1(0) ≃ CP2 is not invariant under �D. Furthermore, the +structure of the foliation �D (and hence that of D) is essentially as complicated as the structure +of the blow-up �F of F, where F denotes the foliation induced by X. This observation deserves +further comments. +To begin with, recall that �C3 can also be seen as the tautological line bundle over π−1(0) ≃ +CP2. The bundle projection will be denoted by �Π : �C3 → π−1(0) since it can naturally be +identified with the canonical projection Π : C3 \ {(0, 0, 0)} → CP2. Next, recall that, unlike +�D, the foliation �F leaves the exceptional divisor invariant. +In particular, at a point p of +π−1(0) ≃ CP2 that is regular for �F, this foliation defines a direction up ∈ Tpπ−1(0). Next, for p +“sufficiently generic”, the leaf of �D intersects transversely π−1(0). This transverse intersection +naturally defines a direction vp ∈ Tpπ−1(0). It is immediate to check that the directions of vp +coincides with the one defined by �F. Denoting by �F|π−1(0) the foliation on π−1(0) obtained by +restriction of �F, we have the following: +Lemma 3.5. The leaves of the foliation �D are of the form �Π−1(L) where L is a leaf of �F|Π−1(0). +Similarly, every leaf of �D is invariant by the foliation �F. +□ +Recalling that every foliation on a projective space is induced by a homogeneous polynomial +vector field, the interest of Lemma 3.5 is actually captured by the following slightly loose state- +ment: every foliation on CP2 is naturally the core foliation for singularities of both dimension 1 +and codimension 1 foliations on (C3, 0). +Before considering some concrete applications of the previous remark, let us close this sec- +tion by pointing out that the above construction allows us to define the core of a dicritical +codimension 1 foliation on (C3, 0) as follows. +Definition 3.6. Let D be a codimension 1 holomorphic foliation defined around the origin of +C3 and assume that the blow up �D of D at the origin does not leave the exceptional divisor + +GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS +11 +π−1(0) invariant. Then the foliation induced on CPn−1 ≃ π−1(0) by the restriction of �F is +called the core foliation of F and its global dynamics is referred to as the core dynamics of F. +3.2. Jouanolou’s example, chaotic dynamics, and their meaning for singularity the- +ory. Let us go back to R. Thom’s question on the existence of separatrices for codimension 1 +foliations on (C3, 0), cf. Definition 2.5. As pointed out in Section 2, it is not always easy to +construct codimension 1 foliations due to Frobenius integrability condition that has to be satis- +fied by the distribution of planes in question. Yet, the discussion revolving around Lemma 3.5 +also indicates a simple way to construct lots of dicritical codimension 1 foliations on C3. More +precisely, every foliation on CP2 yields one such dicritical codimension 1 foliation. +Let then D be a dicritical codimension 1 foliation as above and assume that D admits separa- +trices. Let then S denote a germ of an irreducible separatrix for D. Since S has codimension 1, +there follows the existence of a germ of an irreducible holomorphic function f : (C3, 0) → (C, 0) +such that S coincides with the set {f = 0}. In terms of Taylor series, we set f = �∞ +i≥l fi where +l is the degree of the first non-zero homogeneous component of f. Let C ⊂ CP2 be the curve +defined on the projective plane by the homogeneous equation {fl = 0} (the tangent cone to S). +If we denote by F the core of D (recall than that F is a 1-dimensional foliation on CP2), then +the following can be said: +Lemma 3.7. With the preceding notation, the curve C ⊂ CP2 is invariant by F. +Proof. The foliation D is defined by an integrable 1-form Ω whose Taylor series takes on the +form Ω = �∞ +i=k Ωi where k stands again for the first non-zero homogeneous component of +Ω. A simple argument based on degrees shows that the 1-form Ωk is integrable as well, i.e., +it satisfies Frobenius equation Ωk ∧ dΩk = 0. Similarly, one checks that the (homogeneous) +surface defined by {fl = 0} yields a separatrix for the codimension-1 foliation Dk induced by +Ωk. Set Ωk = F kdx + Gkdy + Hkdz and, as usual, let R denote the radial vector field on C3. +Next recall that a homogeneous vector field of C3 representing F is well defined only up to +a multiplicative constant and addition of a multiple of the radial vector field. Now, since Ωk +is homogeneous, the vector R(p) is contained in the plane defined by the kernel of Ωk(p) at +the point p. Hence, up to eliminating multiplicative factors, a representative vector field X +for F can be obtained by letting X(p) = R(p) ∧ (F k(p), Gk(p), Hk(p)). In particular X(p) lies +in the kernel of Ωk(p), i.e., X is tangent to the foliation Dk. Finally, since at regular points +p ∈ {fl = 0} the tangent space at {fl = 0} and the kernel of Ωk(p) coincide, we conclude that +X is tangent to the surface {fl = 0}. The lemma then follows immediately. +□ +In view of Lemma 3.7, the basic remark of Jouanolou concerning Thom’s conjecture was the +following one: if we can find a foliation F on CP2 leaving invariant no algebraic curve, then the +(dicritical) codimension 1 foliation D arising from combining the radial vector field of C3 and +a representative of homogeneous vector field for F will admit no separatrix. +Jouanolou’s remark is possibly the first instance where the existence of the core dynamics +actually impacts the study of singularities of foliations. From this point of view, the main result +of Jouanolou in [27] can be stated as follows: +Theorem 3.8. [27] For every d ≥ 2, the foliation induced on CP2 by the vector field +Xd = yd∂/∂x + zd∂/∂y + xd∂/∂z +leaves no algebraic curve invariant. +Jouanoulou theorem implies, in particular, that for every fixed d ≥ 2, there exist foliations +of degree d that are not tangent to any algebraic curve of CP2. + +12 +J. REBELO AND H. REIS +Armed with the above theorem, there follows from what precedes that the codimension 1 +Jouanolou foliation Jd, d ≥ 2, of C3 which is defined as the singular foliation spanned by Xd +and the radial vector field is a counterexample to Thom’s question. The well-known explicit +1-form Ω, +Ω = (yxd − zd+1) dx + (zyd − xd+1) dy + (xzd − yd+1) dz , +defining the foliation Jd can promptly be obtained by taking the vector product of Xd and R. +The next question is to wonder how far the core dynamics can influence the study of singu- +larities of foliations, say of dimension 1 on Cn, n ≥ 3. In other words, owing to the discussion +in this section, the detailed understanding of the local structure of one such foliation arguably +passes through the global description of its core foliation. This understanding would require, +in particular, a (global) control of the dynamics of the core foliation. At this point, we might +wonder whether it is possible to obtain such an accurate local description of all 1-dimensional +foliations on, say, (C3, 0). From the standpoint emphasized above, an easier question would be +to provide a reasonable global description of all or nearly all foliations on CP2. Unfortunately, +the latter question does not seem to admit an affirmative answer as it follows from Loray-Rebelo +theorem [28] as stated below. +Fix positive integers n and d, with min{n, d} ≥ 2. A straightforward counting of parameters +shows that the space Fol(d)(n) +CP +of degree d foliations on CPn can be identified with a Zariski-open +set of the complex projective space of dimension +(d + n + 1)(d + n − 1)! +d!(n − 1)! +− 1. +This space of foliation can then be furthered moduled out by the action of the automorphism +group PSL (n + 1, C) of CPn but this will not be needed in the sequel. The main upshot here is +that Fol(d)(n) +CP +can be parameterized by a finite dimensional complex manifold and, in particular, +inherits of a natural topology. With this notation, the main result of [28] reads as follows: +Theorem 3.9. [28] Fixed n, d ≥ 2, there exists a non-empty open subset U ⊂ Fol(d)(n) +CP +such +that every foliation F lying in U satisfies all the conditions below: +(1) All singular points of F are hyperbolic. In particular, they form a finite set. +(2) Minimality: Every leaf of F is dense in CPn. +(3) Ergodicity: Every measurable set of leaves has either zero or total Lebesgue measure. +(4) Rigidity: If F′ ∈ Fol(d)(n) +CP +is conjugate to F by a homeomorphism h : CPn → CPn that is +close to the identity, then F and F′ are also conjugate by an element of PSL (n + 1, C). +The level of dynamical complication exhibited by the foliations indicated above puts any ac- +curate description of them basically out of reach. Moreover, even up to topological conjugation, +it is not possible to achieve a reasonable list of “models” or “normal forms” owing to the above +indicated rigidity phenomenon. +It is convenient to expound a bit on the consequences of Theorem 3.9 from the point of view +of singularity theory for 1-dimensional foliations on dimensions 3 and greater. Consider then a +foliation lying in the set U ⊂ Fol(d)(n) +CP +provided by Theorem 3.9. As a foliation defined on CPn, +it can be represented by some homogeneous polynomial vector field X on Cn+1. In other words, +if F is the foliation on Cn+1 induced by the local orbits of X then �F, its (one-point) blow-up at +the origin, leaves the exceptional divisor π−1(0) ≃ CPn invariant and is such that the restriction +of �F to π−1(0) ≃ CPn is naturally identified with the initial foliation in U ⊂ Fol(d)(n) +CP +. +Now recall that the vector field X is not uniquely defined: most notably, we can add to X +any multiple of the radial vector field by a homogeneous polynomial of degree d − 1. Since, in + +GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS +13 +addition, the singularities of the initial foliation in U are all hyperbolic, it is easy to conclude +that the vector field X can be chosen so as to fulfill the following conditions: +(1) The foliation F has an isolated singularity at the origin of Cn+1. +(2) The foliation �F, viewed as foliation on a manifold of dimension n + 1, still have only +hyperbolic singularities. +Furthermore, a generic choice of the initial foliation in U and of the vector field X allows +us to rule out the existence of resonances at the singular points of �F as well. Thus, all the +singularities of �F are, in fact, linearizable. Also, all the above mentioned characteristic are +stable under higher order perturbations of a representative vector field. The situation can then +be summarized as a statement in itself. +Theorem 3.10. (Corollary of [28]) For every degree d ≥ 2, there exists a non-empty open set +V of homogeneous vector fields of degree d in X(Cn+1, 0) such that every germ of foliation F +represented by a holomorphic vector field X having the form X = Xd +h.o.t., with Xd ∈ V and +where h.o.t. stands for higher order terms, satisfy all of the following conditions: +(1) The one-point blow up �F of F at the origin leaves the exceptional divisor π−1(0) ≃ CPn +invariant. +(2) All singular points of �F are hyperbolic and linearizable. In particular, �F has exactly +dn+1 − 1 +d − 1 +singular points and all of them lie in π−1(0) ≃ CPn. +(3) The restriction of �F to π−1(0) ≃ CPn defines a degree d foliation of CPn lying in the +open set U ⊂ Fol(d)(n) +CP +given by Theorem 3.9. +Let us point out that the formula in item (2) for the number of singular points of �F, i.e., for +a degree d foliation on CPn all of whose singular points are hyperbolic is well known and can be +proved in a variety of ways. For example, by choosing affine coordinates yielding a “hyperplane +at infinity” on which the foliation has no singular point and then applying B´ezout theorem +to the corresponding polynomial vector field representing the foliation in the above indicated +affine coordinates. +Remark 3.11. Naturally the content of Theorem 3.10 can be adapted to germs of codimension 1 +dicritical foliations on (C3, 0). +To close this section, it is convenient to make a parallel with the discussion in Section 2.2 for +singularities of foliations on (C2, 0) so as to better appreciate the difficulties arising from the +existence of wild core dynamics as stated in Proposition 3.10. +(A’) Generalizations of Seidenberg theorem to (Cn, 0). The problem is wide open for +n ≥ 4 though sharp desingularization theorems are now established for n = 3. The topic is +of undisputed interest since virtually all general statements about singularities rely, directly or +indirectly, on a suitable “resolution theorem”. Yet, for n ≥ 3, the ability to obtain a model +of the foliation where all singular points are “simple enough” might still be a long way off of +providing an accurate description of the singularity in question. +To substantiate the above claim, it suffices to consider the local foliations F on (Cn, 0) +provided by Theorem 3.10. The blow-up �F of F at the origin provides a birational model +for F possessing only “simple singular points”: in fact, all singularities of �F are hyperbolic +and linearizable. In other words, the local behavior of �F around each of its singular points +is essentially trivial and promptly available. The very complicated dynamical behavior of F + +14 +J. REBELO AND H. REIS +around the origin is, however, encoded in its core dynamics (cf. Lemma 3.5) but the global +nature of the core dynamics prevents resolution theorems to yield any insight into this dynamical +system. +(B’) Taming the core dynamics. If one is to fully understand the structure of a foliation +around a singular point, then an accurate description of its core dynamics needs to be envisaged. +If Proposition 3.10 tells us this is a kind of unrealistic goal, it also raises the question of +“selecting” those classes of singular points allowing a more detailed description. This is a very +interesting point as it hints at considering the connections between singularity theory and the +remainder of Mathematics or, even, Physics. Singularities playing a special role in problems +from Geometry, Complex Analysis and/or Integrable Systems are likely to be amenable to a +more complete analysis. +Examples of these situations will be discussed in the forthcoming +sections. +In terms of “taming core dynamics”, of course the ideal situation would be to have a core +foliation defining an “integrable system” in some suitable sense. Alternatively, for a number of +problems, it might be enough to ensure the existence of (“sufficiently many”) algebraic invariant +curves. An important issue involving invariant curves is that more often than not the dynamics +of the foliation in question can be investigated in more details on a neighborhood of them, +especially when their fundamental group contains more than a single generator. This study, +whereas of more global nature, is somehow akin to “Mattei-Moussu pseudogroup technique” +mentioned in Section 2.2. Interesting examples where this point of view have successfully been +employed - even outside the scope of singularity theory - include [24], [28], and [19]. +(C’) Dynamics of pseudogroups acting on (Cn, 0). The perspective of focusing in the +local dynamics arising from the holonomy of an invariant algebraic curve in higher dimensions +naturally leads us towards considering the dynamics of subgroups of Diff (Cn, 0), n ≥ 2. As +was to be expected, many new dynamical phenomena arise for n ≥ 2 compared to the situation +n = 1. +As pointed out in Section 2.2 much is known about the dynamics of subgroups of +Diff (C, 0) whereas for subgroups of Diff (Cn, 0), the theory is still in its early stages. +Nonetheless, we mention that generalizations to higher dimensions of Mattei-Moussu’s theo- +rem on groups with finite orbits is by now well understood, see [46], [48], [52]. These results are +likely to have impact in problems about existence of first integrals but they might also provide +insight in higher dimensional versions of the so-called “analytic limit set”, see [3]. +Finally a major issue in the theory is to find sharp conditions to extend to higher dimensions +the Shcherbakov-Nakai theory of local vector fields in the “closure of the group” [53], [54], +[38]. Very little is known about this question aside from some results in [28] which rely on the +existence of a hyperbolic contraction for the group in question. This assumption looking rather +far from sharp, the topic appears to be ripe for significant progress. +4. Resolution theorems in dimension 3 +In the remainder of this survey we will discuss relatively recent progress in some of the several +aspects of singularity theory. This section is devoted to “resolution theorems” while the next +two sections will basically review the general problem of invariant varieties and the study of a +particular and important class of singular points, namely the semicomplete ones. In the course +of these discussions theorems providing - at various degrees - some control on the core dynamics +in question will play a prominent role. +As previously indicated, theorems on reductions of singular points are always of paramount +importance in the theory whether or not there are major difficulties lying out of their reach +(e.g. complicated core dynamics). For foliations defined on complex 2-dimensional manifolds + +GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS +15 +(or varieties), Seidenberg’s theorem provides a sharp reduction of singularities theorem (a.k.a. +“resolution theorem”) that is particularly easy to manipulate. Beyond dimension 2, decisive +results exist only in dimension 3, where it is already necessary to distinguish between foliations +of dimension 1 and foliations of codimension 1. +This section is devoted to reviewing and +explaining the main “resolution theorems” for 1-dimensional foliations in dimension 3. +Owing to the classical Hironaka resolution theorem, we can assume that our singular foliations +are always defined on manifolds (i.e. smooth complex spaces). Furthermore, since the problems +are local, we may assume them to be defined on a neighborhood of the origin of Cn. The case +n = 2 being settled by the above mentioned theorem of Seidenberg, we assume from now on +that n = 3, i.e., our foliations are defined on a neighborhood of the origin of C3. +Working on (C3, 0), we need to distinguish between foliations of dimension 1 and foliations +of codimension 1. The case of codimension 1 foliations was settled earlier in [8]. However, the +story involving foliations of dimension 1 - the main object of this survey - is longer and more +elusive. +Resolution results for foliations of dimension 1 on (C3, 0) have first appeared in [6], where the +author proves the existence of a formal local uniformization theorem. In this work, the author +also hints at the existence of a new phenomenon involving singularities possessing a certain +formal separatrix (i.e. a formal curve invariant by the foliation) which posed serious difficulties +to resolve the singularity by means of standard blow ups. The issue was made clear by Sancho +and Sanz who provided explicit examples of foliations in (C3, 0) that cannot be reduced by +sequences of standard blow-ups centered at sets contained in the singular loci of the initial +foliations and its transforms. +After the examples found by Sancho and Sanz, the next truly major result in the area is +due to D. Panazzolo [39]. In [39], Panazzolo considers singularities of real foliations in (real) +dimension 3. He works in the real setting, rather than in the complex one, mostly due to the +fact that his original motivation lied in Hilbert’s problem about the number of limit cycles of +a polynomial vector field on R2. In his work, Panazzolo shows that the corresponding germs +of foliations can always be turned into a foliation all of whose singular points are elementary +by means of a finite sequence of weighted blow ups centered at singular sets. +The proof is +constructive and actually provides an algorithm to obtain the desired reduction of singularities. +Later, relying on Panazzolo’s algorithm introduced in [39], McQuillan and Panazzolo were able +to provide a very satisfactory answer to the generalization of Seidenberg’s theorem for foliations +on (C3, 0) in [36], [37]. +The preprint [36] was made available in 2007 and a few years later, Cano, Roche, and +Spivakovsky revisited the topic from the point of view of valuation theory, see [9]. Their strategy +is in line with Zariski’s general approach to desingularization problems and, hence, is essentially +divided in two parts. First, for a given foliation, we seek to “simplify” only the singularities +lying in the center of a given valuation (identified with its transforms, or extensions, through +blow-ups). Resolution results for singularities lying in the center of a valuation are often referred +to as local uniformization theorems and the first part of Zariski approach aims at obtaining this +type of statement. Once a convenient local uniformization result is obtained, the second part +of Zariski approach deals with its globalization. More precisely, once it is proved that for every +valuation ν, the singularities lying in the center of ν can be simplified (in some appropriate +sense), we try to conclude that, in fact, all singularities of the foliation can simultaneously be +simplified in the same sense. When it comes to applying this point of view to singularities of +foliations most of the difficulties related to the globalization procedure are handled pretty well +a very general (axiomatic) gluing theorem due to O. Piltant [40]. Owing to Piltant’s theorem, + +16 +J. REBELO AND H. REIS +it is fair to say that the fundamental difficulty of resolution problems for foliations, in arbitrary +dimensions, revolves around obtaining suitable local uniformization theorems. +In view of what precedes, the content of [9] can roughly be summarized by claiming the exis- +tence of a birational model for the initial foliation where the singular points are log-elementary. +The reader is referred to [9] for the definition of log-elementary singular points. For our pur- +poses, it suffices to know that such singularities are, at worst, quadratic in the sense that they +are locally given by a representative vector fields with non-zero second-jet at the singular point +in question. One of the goals of [49] was to complete the work of Cano-Roche-Spivakovsky by +deriving “final models” similar to those of [37], in order to obtain a global resolution theorem +comparable to [36], [37] through Zariski classical approach. +We will compare the resolution theorems for foliations obtained by McQuillan-Panazzolo +in [37] and by ourselves in [49], they correspond to Theorem 2 and Theorem A of the respective +papers. +In particular, it will be seen that in the context of foliations the two results are +pretty much equivalent and can be summarized by the following assertion: given a singular +holomorphic 1-dimensional foliation F on (C3, 0), there exists a birational model of F where +all singular points are elementary. In this sense, the only difference between the theorems in +question will be down to the way in which the desired birational model is constructed. +4.1. Persistent nilpotent singularities. As already mentioned, Sancho and Sanz have showed +the existence of foliations in (C3, 0) that cannot be reduced by sequences of standard blow-ups +with centers contained in the singular set of the initial foliation and its transforms. In fact, +their result is slightly more general in the sense that we may allow for blow-ups of invariant +centers not necessarily contained in the singular locus. As a matter of fact, they have pro- +vided a 3-parameter family of foliations whose elements cannot be turned into a foliation all +of whose singularities are elementary by means of blow ups centered in the singular loci and +whose generic element cannot be turned into a foliation with elementary singular points even +if invariant centers are allowed. This family of foliations is represented by the family of vector +fields Xα,β,λ taking on the form +(3) +Xα,β,λ = x +� +x ∂ +∂x − αy ∂ +∂y − βz ∂ +∂z +� ++ xz ∂ +∂y + (y − λx) ∂ +∂z . +Accordingly, foliations in this family will be denoted by Fα,β,λ. The foliations Fα,β,λ are nilpo- +tent at the origin in the sense that so are the vector fields Xα,β,λ. For reference, it is convenient +to make accurate the notion of nilpotent foliation. +Definition 4.1. A (1-dimensional) holomorphic foliation is said to have a nilpotent singularity +at a singular point p if its representative vector field around p has non-zero nilpotent linear part +at p. +The above notion of nilpotent singularity is well defined since it does not depend on the choice +of the representative vector field. Also, whenever no misunderstanding about the singular point +in question is possible, we will abridge notation by simply saying that F is a nilpotent foliation. +Going back to the nilpotent foliations Fα,β,λ, we note that the plane {x = 0} is invariant by +them and that it contains the singular set of Fα,β,λ which coincides with the axis {x = y = 0}. +Recalling that a singular point is said to be elementary if the representative vector field possesses +at least one eigenvalue different from zero, we now have the following: +Proposition 4.2. The foliations in the family Fα,β,λ cannot be turned into a foliation all of +whose singular points are elementary by means of a sequence of standard blow-ups with centers +contained in singular sets. + +GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS +17 +Sketch of Proof. Consider the one-point blow-up centered at the origin of (C3, 0) and let π +stands for the blow-up map. Let then (x, u, v) be the affine coordinates for the blown-up space +where y = ux and z = vx. The pull-back π∗Xα,β,λ of the vector field Xα,β,λ is given by +π∗Xα,β,λ = x +� +x ∂ +∂x − (α + 1)u ∂ +∂u − (β + 1)v ∂ +∂v +� ++ xv ∂ +∂u + (u − λ) ∂ +∂v , +whose expression is similar to the expression of Xα,β,λ. In fact, the main difference between the +two expressions concerns the last term. Note, however, that the origin of the present coordinates +is not contained in the singular set of the induced foliation, which is given by {x = 0, u = λ}. +Thus, if we consider the translation T(x, u, v) = (x, u + λ, v + µ), the pull-back of π∗Xα,β,λ +through T is given by +x +� +x ∂ +∂x − (α + 1)u ∂ +∂u − (β + 1)v ∂ +∂v +� ++ x(v + µ − λ(α + 1)) ∂ +∂u + (u − µ(β + 1)x) ∂ +∂v . +In the particular, if we choose µ = λ(α + 1), the vector field in question coincides with the +vector field Xα+1,β+1,λ(α+1)(β+1). In other words, the transformed foliation of Fα,β,λ contains +a nilpotent singular point belonging to the (initial) Sancho-Sanz family. +It can be checked +that the same issue occurs if the blow-up centered at the curve of singular points of Fα,β,λ is +considered. +□ +Summarizing what precedes, every sequence of blow ups as above applied to a foliation in +Sancho-Sanz family lead to a foliation having a singular point where the foliation is locally +conjugate to another foliation in the initial family. In particular, all transformed foliations will +exhibit a nilpotent singular point. This nilpotent singular point has a geometric interpretation +naturally related to the issues raised by Cano in [6] for a resolution by standard blow-ups. In +fact, by elaborating in the above indicated argument, Sancho and Sanz have shown that the +parameters α, β, λ can be chosen so that the foliation associated with the vector field Xα,β,λ +possesses a strictly formal separatrix S = S0 through the origin. Moreover, given a sequence of +blow ups as before, the sequence of points {pn} in the exceptional divisors corresponding to the +position of the (persistent) nilpotent singularity is determined by the sequence of transforms +{Sn} of the formal separatrix S = Sn. We should still note that, the fact that every separatrix +Sn is stricty formal says that even in the case we allow blow-ups to be centered at analytic +invariant curves that are not necessarily contained in singular set of the foliation, a resolution +procedure still does not exist. +In terms of the relation between foliations and - possibly formal - separatrices, a natural object +that plays an important role is the notion of multiplicity of the foliation along the separatrix. +Let X be a representative vector field of F and ϕ the Puiseux parametrization of S. Let ϕ∗X|S +stands for the pull-back of the restriction of X to S. If ϕ∗X|S = g(t)∂/∂t, then the multiplicity +of F along S is defined as the order of g at 0 ∈ C (assuming that ϕ(0) coincides with the +singular point). +In [49], we introduced the notion of persistent nilpotent singular point which is as follows. +Definition 4.3. A nilpotent singular point p0 of a foliation F0 is said to be persistent if there +exists a formal separatrix S0 for F0 through p0 such that for every sequence of blowing-ups +F0 +π1 +←− F1 +π2 +←− · · · +πl +←− Fn +where Fi stands for the transformed of Fi−1 through the (standard) blow-up centered at some +Ci−i ⊆ Sing (Fi−1) containing the point pi−1 (selected by the transformed separatrix Si−1, +in the sense that it corresponds to the intersection of Si−1 with the excetional divisor), the +following conditions are satisfied + +18 +J. REBELO AND H. REIS +(a) the singular points pi are all nilpotent singular points for the corresponding foliations; +(b) the multiplicity of Fi along Si does not depend on i. +The multiplicity of a foliation F along a separatrix (possibly a formal one) S is defined as +follows. Let X be a representative vector field of F and ϕ the Puiseux parametrization of S. +Let ϕ∗X|S stands for the pull-back of the restriction of X to S. If ϕ∗X|S = g(t)∂/∂t, then the +multiplicity of F along S, mult(F, S), is defined as the order of g at 0 ∈ C (assuming that ϕ(0) +coincides with the singular point). +The role played by the notion of multiplicity of a foliation along a separatrix is closely related +to its natural behavior under blow ups. Recall that the order of a foliation at a singular point +is nothing but the order of a representative vector field X , i.e. the degree of the first non-zero +jet of X, at the singular point in question. With this notation, assume that �F is obtained by +blowing up F at a singular point p. Assume also that S is a (formal) separatrix of F at p and +denoted by �S the transform of S which yields a (formal) separatrix for �F at the point �p. Then +we have: +(4) +mult( �F, �S) ≤ mult(F, S) +with equality holding if and only if the order of F at p equals 1. The same formula holds for +blowing ups centered at a curve contained in the singular set of F, up to considering a variant +of the notion of “order of the foliation” that is adapted to the center of the blow up, for details +see [49] or the discussion at the end of Section 5.1. +It is easy to understand the interest of the multiplicity of a foliation F along a separatrix +from the above perspective: if the existence of a (formal) separatrix S is ensured, then its +multiplicity will drop strictly providing that the order of F at the singular point in question +is greater than or equal to 2. +Moreover, once this decreasing sequence stabilizes, then the +corresponding singular point is either elementary or nilpotent. From this it also follows that +it is useful to understand persistent nilpotent singular points in order to establish resolution +theorems for foliations. +Clearly, in dimension 2, persistent nilpotent singular points do not exist as follows from +Seidenberg theorem. +In dimension 3, their existence is established by the above discussed +examples due to Sancho and Sanz. A characterization of these points in dimension 3 in terms +of normal forms can be formulated as follows. +Theorem 4.4. [49] Assume that F cannot be resolved by a finite sequence of standard blow- +ups centered at singular sets. Then there exists a sequence of one-point blow ups (centered at +singular points) leading to a foliation F′ with a singular point p around which F is given by a +vector field of the form +(y + zf(x, y, z)) ∂ +∂x + zg(x, y, z) ∂ +∂y + zn ∂ +∂z +for some n ≥ 2 and holomorphic functions f and g of order at least 1 with ∂g/∂x(0, 0, 0) ̸= 0. +Furthermore we have: +(1) The resulting foliation F′ admits a formal separatrix at p which is tangent to the z-axis; +(2) The exceptional divisor is locally contained in the plane {z = 0}. +Theorem 4.4 deserves a couple of comments as it has a natural analogue in [37], namely: +• In [37], the authors obtain an alternative characterization of persistent nilpotent singu- +larities which are presented as singular points arising from elementary ones by means of +a Z/2Z-orbifold singularity, see Section 4.2 for more details. It is relatively straightfor- +ward to establish the equivalence between their characterization and the normal forms +provided by Theorem 4.4. + +GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS +19 +• As in [37], an immediate consequence of the normal forms in Theorem 4.4 is that every +persistent nilpotent singular point can immediately be turned into elementary ones by +means of a single blow-up of weight 2, see [49]. +In closing this section, let us point out the family of vector fields described in Theorem 4.4 is +a genuine extension of the Sancho-Sanz family, albeit one naturally obtained by following their +construction. Indeed, for persistent nilpotent singular points, the multiplicity of the foliation +along the corresponding (formal) separatrix is fully invariant under blow ups whose centers are +contained in singular sets, cf, Formula 4. In the Sancho-Sanz family, all multiplicities are equal +to 2 so that for n ≥ 3, Theorem 4.4 yields examples that cannot be turned in Sancho-Sanz +examples by means of successive blow ups (and conversely). For example, vector fields in the +family +Xλ = (y − λz) ∂ +∂x + zx ∂ +∂y + z3 ∂ +∂z , +with λ ̸= 0, yield foliations Fλ with persistent nilpotent singularities arising from a (strictly) +formal separatrix Sλ. The multiplicity of Fλ along Sλ being equal to 3. +4.2. The desingularization theorem of McQuillan-Panazzolo. The purpose of this para- +graph is to explain in detail the desingularization theorem proved in [37]. As previously men- +tioned, McQuillan and Panazzolo work from the start in the category of weighted blow-ups, +thus not limiting themselves to stardard ones. Unlike standard blow-ups, that keep the smooth +nature of the space, weighted blow-ups lead to singular ambient spaces. Yet, the singularities in +question are of orbifold-type and hence of a rather simple nature. Whereas singular, it should +be pointed, that the ambient space obtained after a sequence of finitely many weighted blow +ups still is birationally equivalent to the initial one. In particular, foliations can be transformed +without any restrictions under weighted blow ups to yield new birational models for them. +Keeping in mind the issues pointed out above, let us summarize the contents of [37]. The +paper [37] is essentially divided into two parts. +Its first part is devoted to prove that the +algorithm of [39] - leading to a resolution of singularities by means of a sequence of weighted +blow-ups for real analytic foliations on (R3, 0) - applies equally well in the general case of +holomorphic foliations on (C3, 0). The algorithm in question thus provides a birational model +for the foliation on a space possessing orbifold-type singular points. Note that, since we are +dealing with (singular) foliations on spaces with singular points of orbifold type, a word is +needed about the meaning of “elementary singular points”. In this regard, the singular point of +the foliation is said to be elementary if it is given by an elementary singular point in a orbifold +coordinate for the space. In particular, there are an open set U ⊂ C3 and finitely ramified +map from U to a neighborhood of the orbifold singular point such that when the foliation is +pulled-back to the open set U ⊂ C3 only elementary singular points are obtained. +In the second part of [37], the authors consider the problem of resolving the singular points +of the ambient space while keeping the singular points of the foliation elementary. They prove +that a resolution for such singularities exists except when the singular point correspond to a +Z/2Z-orbifold. In other words, they have shown that, given a foliation F, it is always possible +to obtain a birational model for F possessing only Z/2Z-orbifold singular points and where all +the singular points of the foliation in question are elementary. In turn, the singularities asso- +ciated with Z/2Z-orbifolds that appear in the end of the previous construction can be turned +into elementary singularities by means of a single blow-up of weight 2. These singularities asso- +ciated with Z/2Z-orbifolds actually correspond to the previously described persistent nilpotent +singular points. +The result in [37] can thus be stated as follows: + +20 +J. REBELO AND H. REIS +Theorem 4.5. [37] Let F be a singular holomorphic foliation on (C3, 0). There is a sequence +of weighted blow-ups +(5) +F0 +π1 +←− F1 +π2 +←− · · · +πl +←− Fl +satisfying the following conditions: +(i) The center of each weighted-blow up is strictly invariant with respect to the quasi- +homogeneous filtration in question. +(ii) The ambient space is an analytic space of dimension 3 whose singular points are Z/2Z- +orbifold type and the total blow-up map π1 ◦ · · · ◦ πl is birational. +(iii) The singular points of Fl are elementary in orbifold coordinates. +Let us close this paragraph with a comment concerning item (i) of Theorem 4.5. Note that +this item is not emphasized in [37] though it is a characteristic property of Panazzolo’s algorithm +in [39]. Whereas, as far as foliations are concerned, this is a minor issue - as it would also be +the case of blow ups centered away from the singular locus (whether or not the blow ups are +weighted) - the issue becomes relevant when our main interest lies in vector fields, rather than +foliations, see Section 4.4 +4.3. Resolution following [49]. In [49], we also establish the existence of a birational model +for F where all singularities of F are elementary except for finitely many ones that can be +turned into elementary singular points by means of a single blow-up of weight 2. To be more +precise, our resolution result for foliations can be stated as follows. +Theorem 4.6. [49] Let F denote a singular holomorphic foliation defined on a neighborhood +of (0, 0, 0) ∈ C3. Then there exists a finite sequence of blow-up maps along with transformed +foliations +(6) +F = F0 +π1 +←− F1 +π2 +←− · · · +πl +←− Fn +satisfying all of the following conditions: +(1) The center of the blow-up map πi is (smooth and) contained in the singular set of Fi−1, +i = 1, . . . , n. +(2) The singularities of Fn are either elementary or persistently nilpotent. +(3) The number of persistently nilpotent singularities of Fn is finite and each of them can +be turned into elementary singular points by performing a single weighted blow-up of +weight 2. +The proof of this theorem has essentially two main ideas. The first one concerns a (personal) +comment by F. Cano claiming that “if a foliation cannot be resolved by standard blow ups, then +there must exist a formal separatrix giving rise to a sequence of infinitely near singular points +that never becomes elementary”. This assertion harks back to his earlier works on resolutions +of 1-dimensional foliations [6] and some important results in this direction can also be found in +[9]. To provide a complete proof of Cano’s assertion was therefore a crucial point in the proof of +Theorem 4.6 and the corresponding result is the content of Proposition 4 in [49]. Interestingly +enough, the argument provided in [49] is rather different from the one envisaged by F. Cano. +With Proposition 4 of [49] in place, the main idea to derive Theorem 4.6 is to argue from +the notion of multiplicity of a foliation along a separatrix, as defined in Subsection 4.1. The +sequence formed by a separatrix and its transforms is decreasing so that it stabilizes after finitely +steps. When the sequence becomes stable, the order of the singular point of the foliation must +be 1. Thus either the singularity has become elementary or we can resort to Theorem 4.4 to +characterize it as a persistent nilpotent singularity, which is necessarily isolated among other +possible persistent nilpotent singular points. Therefore this yields a local uniformization theorem + +GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS +21 +in the sense of Zariski. At this point, O. Piltant “gluing theorem” [40] allows one to conclude +Theorem 4.6. +Recalling that persistent nilpotent singular points are in correspondence with Z/2Z-orbifold +type singular points, the differences between Theorem 4.5 and Theorem 4.6 are down to the +way the corresponding birational models are constructed. Unlike Panazzolo [39], our proof of +Theorem 4.6 does not provide any effective algorithm to resolve singularities. In some problems, +however, it might simplify discussions/arguments by sticking to a single type of blow up, the +standard one, provided that the problem in question requires only a theorem asserting the +existence of a resolution, as opposed to an effective manner to obtain the resolution in question. +4.4. A final comment on transforming vector fields. In close this section, let us point out +a virtue of standard blow ups, as used as in Theorem 4.6, that is also present in Theorem 4.5 +thanks to item (i) in the corresponding statement. This concerns vector fields as opposed to +1-dimensional foliations. +In fact, it is not a foliation but rather some holomorphic vector field that is the object of +primary interestin many problems and applications of singularity theory. +Examples of this +situation are provided in Section 5.1 and throughout Section 6. Naturally a vector field X +gives rise to an 1-dimensional foliation F of which a birational model whose all singular points +are elementary may be useful. +Nonetheless, if the vector field X is the object of primary +interest, then the transforms of X have to be considered as well. At this point, the difference +between vector fields and 1-dimensional foliations is summarized by the following self-evident +statement: the transform of a 1-dimensional holomorphic foliation under a rational map is +another 1-dimensional holomorphic foliation, however, the transform of a holomorphic vector +field under a rational map is, in general, a meromorphic vector field. +When applying resolution theorems for 1-dimensional foliations to the study of vector fields +it is therefore relevant to seek to retain the “good” analytic properties of them as much as +possible (again concrete examples are provided in Sections 5 and 6). For example, if we start +with a holomorphic vector field X, we might hope that its transform at the end of a resolution +procedure still is a holomorphic vector field. In this context, we have: +Claim. The transform of a holomorphic vector field under a resolution procedure as in Theo- +rem 4.5 or in Theorem 4.6 still is a holomorphic vector field. +The claim is clear in the case of Theorem 4.6 as it is a basic fact that the blow-up of a +holomorphic vector field centered at its singular locus is again holomorphic. In fact, for the +blown-up vector field to be holomorphic again it suffices the center of the blow-up to be invariant +by the initial vector field. +In the case of Theorem 4.5 this is not immediate as a weighted blow-up may turn a holomor- +phic vector field into a meromorphic one even if its center is contained in the singular set of the +initial vector field. This is where the condition of having centers that are called strictly invariant +with respect to the quasi-homogeneous filtration in question, as used in [39] and reproduced in +the first part of [37], comes into play (see item (i) in Theorem 4.5). This condition, if slightly +technical, ensures that the transform of holomorphic vector fields remains holomorphic. +5. Invariant analytic sets +The problem of existence of invariant manifolds has always been a central theme in the +theory of dynamical systems. Among the many reasons for this, there is the simple fact that +these invariant manifolds usually provide reductions on the dimension of the corresponding +phase-space. For example, in the general theory of hyperbolic systems, the so-called stable +manifolds are examples of invariant manifolds and, in fact, their existence form a cornerstone + +22 +J. REBELO AND H. REIS +of the hyperbolic theory. The existence of stable manifolds for hyperbolic singular points is a +consequence of the general theory and ensured by the well-known Stable Manifold Theorem. +However, whether or not “stable”, invariant manifolds may fail to exist if the singular point is +no longer hyperbolic. The simplest example is provided by the vector field +X = y ∂ +∂x − x ∂ +∂y +all of whose integral curves are circles around the origin of R2. Clearly there is no invariant +manifold in this case. +The general problem of existence of invariant manifolds may also be considered in the context +of holomorphic dynamics. In this case, we look for invariant complex-analytic objects, which +is a much stronger regularity condition. We allow, however, these objects to be singular in the +sense of analytic sets. In other words, we look for invariant varieties, as opposed to actual +manifolds. In the sequel, the word “manifold” will be saved for smooth objects. +As mentioned in section 2, Briot and Bouquet were the first to consider the problem of +existence of separatrices for holomorphic vector fields defined on a neighborhood of the origin +of C2 in [2]. +However, they were not able to establish the existence of separatrices for all +holomorphic vector fields on (C2, 0). This question was settled only much later by Camacho +and Sad in their remarkable paper [5] where the following is proved: +Theorem 5.1. [5] Let F be a singular holomorphic foliation defined on a neighborhood of the +origin of C2. Then there exists an analytic invariant curve passing through (0, 0) and invariant +by F. +Theorem 5.1 is well worth a few additional comments, namely: +• It is somehow surprising that separatrices for holomorphic vector fields on (C2, 0) always +exist despite the much stronger regularity condition for the invariant curve. For example, +for the holomorphic vector field y ∂ +∂x − x ∂ +∂y defined on (C2, 0), the separatrices are given +by the two complex lines y = ±ix and hence are totally contained in the non-real part +of C2 (bar the singular point itself). +• However, as mentioned, we do not require the separatrices to be smooth invariant curves +otherwise no general existence statement would hold. +Indeed, as a simple example, +consider the holomorphic vector field 2y∂/∂x + x3∂/∂y on (C2, 0). Since this vector +field admits f(x, y) = x3 − y2 as first integral, it immediately follows that the only +separatrix of X is the cusp of equation {x3 − y2 = 0}, which is clearly not smooth +at the origin. Fortunately, allowing separatrices to be singular is not a problem, since +Hironaka’s theorem can always be used to desingularize them. +• Also it is important to emphasize that Theorem 5.1 applies only to foliations defined +on smooth ambients. +In fact, if germs of foliations defined on singular surfaces are +considered, then separatrices may fail to exist as shown by Camacho in [4]. +The existence of separatrices is, however, no longer a general phenomenon in dimension 3, +regardless of the dimension of the foliation. As already said, a first example of codimension 1 +foliation on (C3, 0) without separatrices was provided by Jouanolou in [26]. Jouanolou’s ex- +ample essentially hinging from the core dynamics of (dicritical) foliations on (C3, 0), the same +idea enables us to construct plenty of additional examples of codimension 1 foliations without +separatrices (cf. Section 3.2 or the summary below). +Concerning 1-dimensional foliations, examples of foliations without separatrices in dimen- +sion 3 were found by Gomez-Mont and Luengo, [15]. Their work will be discussed in Section 5.2. +For the time being, we will focus on the problem of invariant manifolds for codimension 1 foli- +ations. + +GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS +23 +5.1. Separatrices for codimension 1 foliations induced by pairs of commuting vector +fields. Let us begin by recalling/summarizing the discussion in Section 3.2 where it was shown +how Jouanolou’s method can be used to produce many examples of codimension 1 foliations +without separatrix on (C3, 0). This is as follows. +(i) Every homogeneous polynomial vector field X on C3 that is not a multiple of the Radial +vector field induces a foliation on CP2 corresponding to the so-called core foliation +associated with X. Conversely, given a foliation on CP2, there exists a homogeneous +polynomial vector field on C3 whose core foliation is the given one. +(ii) Let X be a homogeneous vector fields distinct from a multiple of the Radial vector field +R. There follows from the Euler relation that X and R generates a Lie algebra isomor- +phic to the Lie algebra of the affine group. In particular, the distribution generated by +X and R can be integrated to yield a dicritical codimension 1 foliation D. Furthermore, +the core foliation associated with D coincides with the core foliation associated with X. +(iii) Finally, for every fixed degree, Theorem 3.9 ensures the existence of a (non-empty) open +set of foliations on CP2 such that all leaves of each foliation F in this set are dense. In +particular, no foliation in this set admits algebraic invariant curves. The codimension 1 +foliations generated by R and by the homogeneous vector field having one such foliation +as core foliation has no seraparatrix. +In view of what precedes, it is natural to wonder if all example of codimension 1 foliations +without separatrices are among the dicritical ones. In ambient spaces of dimension 3 this, in +fact, holds as proved by Cano and Cerveau in [7]. Their result can be stated as follows. +Theorem 5.2. [7] Let D be a germ of a holomorphic singular codimension 1 foliation on (C3, 0). +If D is not dicritical, then it admits a separatrix. +The proof of the previous result relies heavily on a resolution theorem for non-dicritical +codimension 1 foliations obtained by the authors in the same paper. Note, however, that the +non-dicritical assumption, implies that the transforms of the initial codimension 1 foliation +leave every irreducible component of the exceptional divisor invariant. In other words, away +from singular points, every irreducible component of the exceptional divisor is a leaf of the +foliations in question. This rules out the existence of meaningful core dynamics and making +the problem very much comparable to the 2-dimensional situation handled by Camacho and +Sad in [5] which has a more geometric nature. +In a different direction, experts including F. Cano, D. Cerveau, and L. Stolovitch have since +long wondered what would be the “correct generalization” of Camacho-Sad theorem for (C3, 0), +already at level of codimension 1 foliations. In particular, the idea that a codimension 1 foliation +spanned by a pair of commuting vector fields (not everywhere parallel) might necessarily admit +separatrices was advanced. The question is settled by the theorem below which confirms their +intuition. +Theorem 5.3. [44] Consider holomorphic vector fields X, Y defined on a neighborhood of the +origin of C3. Suppose that they commute and are linearly independent at generic points (so that +they span a codimension 1 foliation denoted by D). Then D possesses a separatrix. +The remainder of this paragraph is devoted to single out a few issues involved in the proof +of Theorem 5.3. This illustrates several points made in the preceding sections, including the +usefulness of “taming” core dynamics (and how “symmetries” may be exploited to this effect) +and the role of resolutions theorems. Concerning the latter, the argument will also highlight +the the importance of having actual vector fields - rather than mere foliations - being “nicely” +transformed during the resolution procedure. + +24 +J. REBELO AND H. REIS +The first ingredient in the proof of Theorem 5.3 is therefore a general resolution of singularities +theorem for codimension 1 foliations in dimension 3. Compared to Theorem 5.2, the main result +in [7] is arguably a theorem of reduction of singularities for the foliations in question under the +additional condition that the foliation should be non-dicritical. Fortunately, Cano has obtained +in [8] a general resolution theorem for codimension 1 foliations on (C3, 0) which applies equally +well to dicritical foliations. +Armed with Cano’s theorem [8], we see that the basic obstacle for the existence of separa- +trices lies in the core dynamics by means of the phenomenon already pointed out in Jouanolou +examples, cf. Section 3. The central point in the proof of Theorem 5.3 is therefore to “tame” +the core dynamics arising from dicritical divisors the resolution procedure applied to D will +have “plenty of algebraic curves”. In the sequel, we shall indicate some simple ideas used to +show that the mentioned core dynamics cannot be “too wild”. +Let us consider the simplest case where we want to blow-up the origin (a degenerate singular +point of D). The first lemma shows that at least one between the vector fields X and Y have +to induce a foliation on the resulting exceptional divisor, unless we have a truly very special +situation that is essentially “linear” (and hence easy to handle). +Recalling that D is spanned by the commuting vector fields X and Y , let FX (resp. FY ) +denote the 1-dimensional singular foliation associated with X (resp. Y ). +Lemma 5.4. Assume that the first jet of both X and Y at the origin are equal to zero. Then +none of the foliations FX or FY is dicritical for the blow-up π of C3 at the origin. +Proof. Denote by � +X and �Y the blow-ups of X and Y at the origin. Similarly, �Fx and �FY will +stand for the blow ups of the foliations FX and FY . Since the vector fields X and Y have zero +linear part at the origin, there follows that both � +X and �Y vanish identically over the exceptional +divisor π−1(0) ≃ CP2. Now assume that, say, X is dicritical for π. Then the leaf of �FX is regular +and transverse to π−1(0) at generic points of π−1(0). Therefore, around one such point, we can +choose local coordinates (u, v, w) such that {u = 0} ⊂ π−1(0) and where of �FX is represented by +the (regular) vector field ∂/∂u. In particular the blow-up � +X takes on the form f(u, v, w)∂/∂u +where f is a holomorphic function (divisible by u). In these coordinates, let the blow-up �Y be +given by �Y = f1∂/∂u+f2∂/∂v +f3∂/∂w. Since [ � +X, �Y ] = 0, there follows that f2 and f3 do not +depend on the variable u. However, these functions must vanish identically for u = 0 since �Y +vanishes identically over π−1(0) (locally given by {u = 0}). Thus they must vanish identically +over an open set and this contradicts the fact that X and Y span a codimension 1 foliation. +□ +Remark 5.5. The argument above shows the importance of transforming vector fields, as +opposed to foliations, in certain cases. In fact, the proof of Lemma 5.4 hinges from the fact +that the transform of the vector field Y vanishes identically over the exceptional divisor π−1(0) +- something that does not make sense for a foliation since the singular set of the latter has +codimension at least 2. +Along similar lines, to ensure that the transformed vector field �Y vanishes identically over +π−1(0), the fact that the origin (center of the blow up) is contained in the singular set of FX (or +more generally, the center of the blow up is invariant under the foliation) was implicitly used. +This is in line with the discussion in Section 4.4. It is often important that the transformed +vector field retains its holomorphic character. +In addition, in quite a few cases, it is also +important that the zero-divisor of the transformed vector field contains all components of the +exceptional divisor arising from the resolution procedure. +Plenty of additional examples of this issue can be found in the theory of semicomplete vector +fields, see for example [13], [18], or [19]. + +GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS +25 +Lemma 5.4 shows that both FX or FY must induce a foliation on CP2. If the foliations +induced are different, then it follows from the discussion in Section 3 that π−1(0) is invariant +by D. Hence we can assume that they do coincide. Recalling that the order of a vector field +at a singular point p is nothing but the degree of the first non-zero homogeneous component of +its Taylor series based at the point in question, the preceding implies: +Lemma 5.6. The orders at the origin of X and Y can be assumed to be different. +Proof. Assume that X and Y have the same order at the origin. Because they induce the same +foliation on CP2, they will differ by a multiple of the radial vector field (up to multiplying, say +X, by a non-zero constant). Hence, by considering Z = X − Y , there follows that the foliation +D is still spanned by X and Z and the we still have [X, Z] = 0. This is however impossible +since Z is clearly dicritical so that a contradiction with Lemma 5.4 arises at once. +□ +Denote by XH (resp. Y H) the first non-zero homogeneous component of X (resp. Y ) at the +origin. Owing to the above lemma, we can assume that the degree of Y H is strictly greater than +the degree of XH. The preceding implies that the core dynamics of either XH, Y H coincides +with the core dynamics of the dicritical foliation D. Now the following proposition provides +some serious control on the core dynamics in question and, along with its analogue for blow +ups centered at curves, constitutes a fundamental starting point of the discussion conducted in +[44]. +Proposition 5.7. The vector field XH admits a non-constant meromorphic/holomorphic first +integral. +Proof. Owing again to the discussion in Section 3, the dicritical nature of D ensures the existence +of holomorphic functions f and g such that +(7) +fX + gY = Z , +with Z being a holomorphic vector field whose first non-zero homogeneous component at the +origin is a multiple of the radial vector field R. Denoting by f H, gH the first non-zero homoge- +neous components of f, g, there follows from the preceding that f HXH and gHY H must have +the same degree. Furthermore, we have a homogeneous equation +(8) +f HXH + gHY H = hHR +where hH is a homogeneous polynomial - possibly identically zero. In the sequel we assume +that hH does not vanish identically since it is easy to adapt the discussion below to cover this +case as well. +Because X, Y commute, so do XH, Y H. Thus we have +[XH, Y H] += +� +XH, hH +gH R − f H +gH XH +� += +� +XH. +�hH +gH +�� +R − hH +gH [R, XH] − +� +XH. +�f H +gH +�� +XH += +� +XH. +�hH +gH +�� +R − +� +(d − 1)hH +gH − XH. +�f H +gH +�� +XH += +0 +where d stands for the degree of XH. In particular +� +XH. +�hH +gH +�� +R = +� +(d − 1)hH +gH − XH. +�f H +gH +�� +XH . + +26 +J. REBELO AND H. REIS +The expression between brackets on the left hand side (i.e. the expression multiplying R) must +vanish identically for otherwise XH would be a multiple of the Radial vector field R. It then +follows that +XH. +�hH +gH +� += 0 . +In other words, hH/gH is a meromorphic (possibly holomorphic) first integral for XH. +It only remains to prove that hH/gH is not constant. However, if this function is constant +(different from zero since hK does not vanish identically), then we can assume hH/gH = 1 +without loss of generality. Hence dividing (8) by gH, it would follow +f H +gH XH + Y H = R. +This last equation is however impossible since Y H has degree at least 2 and the expression +f HXH/gH is homogeneous. +Therefore hH/gH cannot be constant. +Since the argument is +symmetric in the vector fields X, Y , the last assertion completes our proof. +□ +The key to prove Theorem 5.3 is to observe that the core dynamics of dicritical components +of a foliation like D must leave invariant certain algebraic curves. Clearly Proposition 5.7 along +with some refinements play a role in this proof. +However, it is also clear that the discussion leading to Proposition 5.7 depends heavily on +the vanishing assumption for the first jet of X, Y at the origin. This issue requires to consider +separately some special situations that are referred to as “linear foliations” in the sense that +there is a non-zero first jet involved. Not surprisingly, “linear foliations” can be dealt with +through rather direct methods. +Finally, as it is inevitable in dimension 3, every desingularization procedure requires two types +of blow-ups: beyond blow-ups centered at points, blow-ups centered at curves are needed as +well. In particular, another basic ingredient in the proof of Theorem 5.3 will be analogues, both +in “linear” and “non-linear” settings, of the previous results. This issue has already appeared +in Section 4 (cf. the discussion about Equation 4) and can easily lead to misunderstandings +so that it seems convenient to close this paragraph by carefully explaining the appropriate +formulations. +The following example was pointed out to us by D. Cerveau. It helps to explain the notion +of “zero first jet” in the case of blow-ups centered at smooth curves. The example also high- +lights difficulties related to the existence of first integrals (cf. Proposition 5.7 and a few other +intermediary results used in [44] and not explicitly mentioned here). +Example 1. Consider the pair of vector fields X, Y given by +X = zy ∂ +∂y + z2 ∂ +∂z +and +Y = x2 ∂ +∂x + axy ∂ +∂y +which are quadratic at the origin. Note that these two vector fields commute so that they span +a codimension 1 foliation denoted by D. +The axis {y = z = 0} is invariant by both X and Y . This axis is also contained in the singular +set of D. Let us then consider the blow-up of C3 centered at the axis {y = z = 0} along with +the corresponding transforms of D, X, and Y . It is immediate to check D is dicritical for the +blow up in question. Similarly the foliation FX associated with X is also dicritical for this blow +up which might lead to some confusion with Lemma 5.4. +The explanation for this example lies in the fact that the vector field Y is regular (non-zero) +at generic points of the axis {y = z = 0}. Similarly, its transform under the previous blow +up is regular at generic points of the exceptional divisor. In other words, this case must be + +GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS +27 +considered as a “linear one” and the order of Y with respect to this blow up must, indeed, be +equal to zero. An adequate definition of the order of a vector field with respect to the center of +a blow up is included below. +Let us then provide an accurate definition of order of a vector field when a curve, as opposed +to a single point, is blown-up. To explain the idea, consider first a holomorphic vector field X +with a singular point at the origin along with the corresponding Taylor series. The order of X +at the origin is said to be the degree of the first non-zero homogeneous component of its Taylor +expansion. This can also be viewed as the integer d for which the limit +lim +λ→0 +1 +λd−1 Γ∗ +λX +yields a (non identically zero) holomorphic vector field. Here Γ∗ +λX stands for the pull-back of +X by the homothety Γλ : (x, y, z) �→ (λx, λy, λz). Note that the limit above corresponds to +the first non-zero homogeneous component of the Taylor’s expansion of X at the origin. Next, +assume now that C = {y = z = 0} is contained in the singular set of X so that the blow-up +centered along this curve of singular points will be considered. The order of X with respect to +C is defined as the integer d for which +lim +λ→0 +1 +λd−1 Λ∗ +λX +is a (non identically zero) holomorphic vector field, where Λ∗ +λX denotes the pull-back of X by +the homothety Λλ : (x, y, z) �→ (x, λy, λz). The limit above, for the appropriate choice of d, is +said to be the first non-zero homogeneous component of X with respect to the variables x, y. +In general, the cases in [44] that are called linear are those cases in which the vector field has +order 1 or zero, with respect to the center of the blow up in question. In particular, with the +above definition, it can immediately be checked that the vector field Y of Example 1 has order +zero with respect to C = {y = z = 0}, although its order at the origin is 2. +5.2. Separatrices for foliations of dimension 1. This paragraph is devoted to discussing +in detail the problem about existence of separatrices for 1-dimensional foliations. Contrasting +with the case of codimension 1 foliations, it will soon be seen that the influence of core dynamics +in the existence of these separatrices is rather limited. In fact, the existence of separatrices for +foliations of dimension 1 is an phenomenon having, in a suitable sense, a very local nature: it +essentially hinges from two basic ingredients, namely: +• The analysis of simple singularities which is basically conducted by direct methods +involving normal forms and divergent series. +• Geometric considerations involving the relative positions of the simple singularities in +question. +In this regard, and provided that a convenient resolution of singularities theorem is available, +the problem somehow retains the same nature regardless of the dimension. More precisely, the +difficulties arising from increasing the dimension stem either from the evident fact that simple +singularities are not always easy to describe (e.g. saddle-nodes of high codimension) and from +the fact that the number of possible arrangements of their relative positions increase as well. +As previously said, after Camacho-Sad theorem in [5] establishing the existence of separatrices +for every foliation on (C2, 0), Gomez-Mont and Luengo found a foliation on (C3, 0) that admits +no separatrix. Let us begin by providing an outline of their construction. +5.3. On Gomez-Mont and Luengo counterexample. Their example of foliation without +separatrix on (C3, 0) relies on two simple remarks. Consider a foliation F on (C3, 0) given by +a holomorphic vector field satisfying the following conditions + +28 +J. REBELO AND H. REIS +(1) The origin (0, 0, 0) ∈ C3 is an isolated singularity of X +(2) J1X(0, 0, 0) = 0 but J2X(0, 0, 0) ̸= 0, where JkX(0, 0, 0) stands for the jet of order k +of X at the origin (k = 1, 2). +(3) The quadratic part X2 of X at (0, 0, 0) is a vector field whose singular set has codimen- +sion 2. Also X2 is not a multiple of the Radial vector field x∂/∂x + y∂/∂y + z∂/∂z. +Assume that F has a separatrix C and consider the blow-up �F of F centered at the origin. +Denote by π the blow-up map so that �F = π∗F and let π−1(0) denote the exceptional divisor +isomorphic to CP2. Since X2 is not a multiple of the Radial vector field, there follows that +π−1(0) is invariant by � +F so that the restriction of �F to π−1(0) can be seen as a foliation of +degree 2 on CP2 (cf. item (3)). +Because π−1(0) is invariant by �F, the transform π−1(C) of the separatrix C can only intersect +π−1(0) at singular points of �F. Furthermore, all of these singular points lie in π−1(0) since X +has an isolated singularity at the origin. In other words, π−1(C) must be a separatrix (not +contained in π−1(0)) for one of the singular points of �F. +Now, the second ingredient is as follows: as a foliation of degree 2 on CP2, the restriction +�F|π−1(0) of �F to π−1(0) has at most (and generically) 7 singular points. Since it is hard to +control the position of 7 points in CP2, the authors of [15] started from a foliation satisfying +the following conditions: +(A) The foliation of degree 2 has only 3 singular points (we can think of the foliation as +obtained by letting some of the 7 singular points of a generic quadratic foliation to +“collide in groups”). Naturally the position of 3 points in CP2 can easily be controlled. +(B) Each of the 3 singular points will have an eigenvalue equal to zero in the direction +transverse to π−1(0). The 3 singular points are therefore saddle-node singularities (in +dimension 3). +(C) Furthermore, the authors arrange for the saddle-node singularities to have two equal +(and non-zero) eigenvalues tangent to π−1(0). In other words, the singular points in +question are (codimension 1) resonant saddle-nodes with weak direction transverse to +π−1(0). +(D) As is well known, it is easy to produce examples of codimension 1 saddle-nodes all of +whose separatrices are included in an invariant (2-dimensional) plane tangent to the +directions of the non-zero eigenvalues. +The remainder of the proof in [15] consists of showing that it is, indeed, possible to prescribe +a quadratic X2 and a cubic X3 homogeneous components for the vector field X so as to satisfy +all of the preceding conditions. In this respect, note that conditions (A), (B), and (C) depend +only on the quadratic part X2. The role played by the appropriately chosen cubic parte X3 +can be summarized as follows. +• it ensures that each of the singular points of �F are isolated, hence coinciding with the +corresponding singular points of �F|π−1(0). Here the reader may note that the homoge- +neous foliation associated with X2 has singularities all along the fibers of �C3 → �π−1(0) +sitting over the singular points of �F|π−1(0). A higher order perturbation of X2 is thus +needed to provide isolated singular points for the blown-up foliation. +• having ensured the singular points are isolated, the cubic part X3 of X also takes care +of condition (D) +As mentioned, the verification that all these conditions are compatible is conducted in [15] +with the assistance of suitable software to deal with formal computations. + +GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS +29 +5.4. Vector fields and 2-dimensional Lie algebras. In [44], codimension 1 foliations spanned +by pairs of commuting vector fields were considered and it was shown that this condition im- +poses strong constraints on the core dynamics of dicritical components of the codimension 1 +foliation in question. In particular, these constraints have proved to be strong enough to yield +the existence of separatrices for the foliation in question. +In view of the preceding, it was natural to wonder if the 1-dimensional foliations arising from +the vector fields in question would have separatrices themselves. While the answer turned out +to be affirmative, the assumption of having two commuting vector fields can be weakened to +encompass also the case of pairs of vector fields generating the Lie algebra of the affine group. +In fact, the following theorem was proved in [47]: +Theorem 5.8. [47] Let X and Y be two holomorphic vector fields defined on a neighborhood +U of (0, 0, 0) ∈ C3 which are not linearly dependent on all of U. Suppose that X and Y vanish +at the origin and that one of the following conditions holds: +• [X, Y ] = 0; +• [X, Y ] = c Y , for a certain c ∈ C∗. +Then there exists a germ of analytic curve C ⊂ C3 passing through the origin and simultaneously +invariant under X and Y . +The theorem above deserves a few additional comments. +• First, the fact that Theorem 5.8 applies to pair of vector fields generating the Lie algebra +of the affine group is in stark contrast with the analogous problem for codimension 1 +foliations. Indeed, every homogeneous vector field of degree at least 2 together with the +radial vector field generate the Lie algebra of the affine group. In particular, Jouanolou’s +and similar examples of codimension 1 foliation without separatrices arise from pairs of +vector fields generating the Lie algebra of the affine group. +• Whereas theorems asserting the existence of separatrices for foliations of dimension 1 +holds interest in their own right, they also have non-trivial applications in the general +problem of understanding globally defined holomorphic vector fields on compact com- +plex manifolds, see Section 6.2. In particular, the paper [47] also includes a non-trivial +applications of Theorem 5.8 in this direction. +• Finally, a relatively minor but yet subtle issue that is worth pointing out is that Theo- +rem 5.8 claims that X and Y possess a common invariant curve without asserting that +the curve in question is invariant by the foliations associated with X and Y . To further +clarify the issue, it is enough to think of the 2-dimensional vector field x∂/∂x: the axis +{x = 0} is invariant by the vector field but does not constitute a separatrix for the +associated foliation. In turn, it might be asked if the foliations associated with X and +Y share an actual separatrix, possibly enlarging the notion of “separatrix” to include +curves fully constituted by singular points of the corresponding foliation. In particular, +it is easy to check that the existence of “common separatrices” always holds when X +is a homogeneous vector field and Y is the radial vector field. Indeed, in this case the +leaves of FY are simply the radial lines. Concerning FX, since it is not a multiple of +the Radial vector field, it induces a 1-dimensional foliation on CP2 by means of the one- +point blow-up of C3 at the origin. The foliation in question possesses isolated singular +points and it can easily be checked that the radial line naturally associated with any +of these singular points is invariant by FX as well. We believe that the existence of a +common separatrix for FX and FY in the general case can also be established. +To finish the section, let us provide an outline of the proof of Theorem 5.8. + +30 +J. REBELO AND H. REIS +Sketch of Proof of Theorem 5.8. Recall that the foliation associated with X (resp. Y ) is de- +noted by FX (resp. FY ). Let D denote the codimension 1 foliation spanned by X and Y . We +have that codim (Sing (D)) ≥ 2. In other words, Sing (D) is of one of the following types: the +union of a finite number of irreducible curves, a single point (the origin), or simply empty (i.e. +D is regular). Since Sing (D) is naturally invariant by X and by Y , the result immediately +holds if dim (Sing (D)) = 1. Hence we can assume without loss of generality that Sing (D) has +codimension at least 3. In other words, either Sing (D) is reduced to the origin or it is, in fact, +empty. +Since the singular set of D has codimension at least 3, Malgrange Theorem [29] implies +that D possesses a non-constant holomorphic first integral f. Let then S = f −1(0) so that +S is an invariant surface for D, i.e. the irreducible components of S are separatrices for D. +In particular, S is invariant by both X and Y . +Next, note that S can be assumed to be +irreducible. Otherwise, the intersection of any two irreducible components of S yields a curve +invariant under both X, Y and the conclusion holds. The surface S can then be assumed either +regular or having an isolated singularity at the origin (again if S contains a curve of singular +points this curve must be invariant by X and Y ). At this point, a couple of remarks are in +order: +• In the case where S is smooth, each of the foliations FX and FY possesses separatrices +owing to Camacho-Sad Theorem [5]. Still it remains to check that these foliations share +a common separatrix. +• As previously mentioned, in the case of singular surfaces, there are examples of foliations +without separatrix (cf. [4] or [15]). This phenomenon needs thus to be ruled out in the +present case. +In general, we proceed as follows. Consider the restrictions of X and Y to S along with the +corresponding tangency locus. This tangency locus is not empty since both X and Y vanish at +the origin. Since the tangency locus Tang (X|S, Y |S) is invariant by both X and Y , the result +immediately holds in the case where its dimension equals 1. So, we shall consider separately +the case where Tang (X|S, Y |S) = {(0, 0, 0)} and the case where Tang (X|S, Y |S) = S. +Assuming that Tang (X|S, Y |S) is reduced to the origin. Then S is a surface with singular +set of codimension at least 2 and equipped with two vector fields that are linearly independent +away from this an analytic set of codimension 2 or greater. This implies that tangent sheaf to +S is locally trivial which, in turn, implies that S is smooth since S is a hypersurface in C3. +However, being smooth, S is locally equivalent to C2 and the tangency locus of two vector fields +cannot be reduced to a single point. The resulting contradiction rules out this case. +Assume now that Tang (X|S, Y |S) = S, i.e. the restrictions to S of X and Y coincide up +to a multiplicative function (defined on S). The existence of the desired common separatrix +is then ensured in the case where S is smooth by Camacho-Sad theorem. It only remains to +consider the case where S has an isolated singular point at the origin. The argument in this case +relies on proving that the (1-dimensional) foliation induced on S by either X or Y possesses +a non-constant holomorphic first integral. The level curve of this first integral containing the +origin then yields the desired separatrix. Details can be found in [47]. +□ +6. Semicomplete vector fields, automorphism groups, and separatrices +The object of this last section is a distinguished class of singularities of vector fields, namely +the semicomplete (singularities of) vector fields. Understanding this class of vector fields, both +at global level and at level of germs, is a problem with interesting applications. As an example +of application, we will see in Section 6.2 that results on singularities of semicomplete vector +fields yield insight in some problems about bounds for the dimension of automorphism group of + +GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS +31 +compact complex manifolds. Another motivation to study these vector fields and their singular +points stems from the very fact that the semicomplete property is somehow akin to the Painlev´e +property for differential equations, albeit the two notions are not equivalent. As a matter of +fact, as it happens with Painlev´e property, semicomplete vector fields are also largely present - +sometimes implicitly - in the literature of Mathematical Physics. +The notion of semicomplete singularity was introduced in [41]. The idea begins with the +definition of semicomplete vector fields on general open sets which is as follows. +Definition 6.1. [41] A holomorphic vector field X defined on an open set U of some complex +manifold M is said to be semicomplete (on U) if for every p ∈ U there exists a connected +domain Vp ⊂ C, with 0 ∈ Vp, and a map φp : Vp → U satisfying the following conditions: +• φp(0) = p +• φ′ +p(T) = X(φp(T)), for every T ∈ Vp. +• For every sequence {Ti} ⊂ Vp such that limi→∞ Ti = ˆT ∈ ∂Vp the sequence {φp(Ti)} +escapes from every compact subset of U. +The third condition in Definition 6.1 basically means that φp : Vp → U is a maximal solution +of X in a sense similar to the notion of “maximal solutions” commonly used for real vector field +and/or differential equations. In this sense, the definition is equivalent to saying that a vector +field is semicomplete if for every p ∈ U the integral curve φ satisfying φ(0) = p has a maximal +domain of definition in C. Closely connected to the notion of maximal domain of definition in +C, we can think of a local integral curve for a vector field X and then extending it over paths +which is always possible as long as we stay in the domain of definition of X. The vector field X +is then semicomplete if these extensions do not give rise to any monodromy and hence can be +merged together in a single (univalued) solution for X which is naturally defined on a maximal +domain in C. +Though global in essence, the above definition has also a local character that is singled out +by the following assertion: if a vector field X is semicomplete on U, then the restriction of +X to every subset V of U is semicomplete as well. Thus the notion of germ of semicomplete +vector field, and hence of semicomplete singularity, makes sense. Furthermore, even at level +of germs, the condition of being semicomplete is far from trivial and, in fact, imposes strong +constraints on the singular points of vector fields as pointed out in [41]. As a matter of fact, +since its introduction, semicomplete singularities have proved time and again that they capture +almost all of the “intrinsic nature” of germs of vector fields admitting actual global realizations +as complete vector fields. +Germs of holomorphic semicomplete vector fields on (C2, 0) were classified by Ghys and +Rebelo in the papers [41] and [13]. In particular, all these vector fields admit a non-constant +holomorphic/meromorphic first integral so that the dynamics associated with them is rather +simple. +After this brief introduction to semicomplete vector fields, the remainder of the section will +focus on two fundamental questions related to them. The first question was somehow motivated +by the results of Ghys and Rebelo in dimension 2 and asks the extent to which the condition +of being semicomplete may tame the core dynamics of the corresponding foliation. In other +words, we ask: +• Are there semicomplete vector fields exhibiting a genuinely complicated core dynamics? +The second question was raised by E. Ghys long ago and, roughly speaking, involves deciding +“how degenerate” can semicomplete singular points be. A prototypical question along these +lines concerns semicomplete vector fields with isolated singular points and can be formulated +as follows: + +32 +J. REBELO AND H. REIS +• Is it true that the second jet of a semicomplete vector field at an isolated singular point is +necessarily different from zero? +This question is affirmatively answered in dimension 2 in the mentioned works by Ghys and +Rebelo. It remains open in higher dimension, though a number of partial results are available +in dimension 3. +Whereas the interest in “taming” the core dynamics associated to singularities of vector +fields has already been emphasized, let us also point out that the general question raised by E. +Ghys has applications to problems about bounds for the dimension of automorphism group of +compact complex manifolds. This issue will further be discussed in Section 6.2. For the time +being, we will focus on the dynamics associated with semicomplete singularities. +6.1. Semicomplete vector fields with complicated dynamics - Guillot’s work [17]. As +previously mentioned, singularities of semicomplete vector fields have very simple dynamics in +complex dimension 2. In fact, even the global behavior of semicomplete vector fields is amenable +to detailed analysis, see [19], [18]. However, this is no longer the case in dimension 3 as follows +from Guillot’s deep work on Halphen vector fields. +This paragraph is basically devoted to +summarizing the main dynamical issues appearing in semicomplete Halphen vector fields while +referring to [17] for a more comprehensive discussion. +Halphen vector fields were first considered by Halphen himself [20], [21]. Apart from his +contribution, let us make clear that all remaining results in this paragraph are due to Guillot +and can be found in [17]. Up to linear equivalence, Halphen vector fields form a three parameters +family of homogeneous polynomial vector fields of degree 2 on C3 explicitly described as +X = +� +α1z2 +1 + (1 − α1)(z1z2 + z1z3 − z2z3) +� ∂ +∂z1 ++ +(9) +� +α2z2 +2 + (1 − α2)(z1z2 − z1z3 + z2z3) +� ∂ +∂z2 ++ +� +α3z2 +3 + (1 − α3)(−z1z2 + z1z3 + z2z3) +� ∂ +∂z3 +An alternate definition pointed out in [17] which already sheds some light in the intrinsic nature +of these vector fields is as follows. +Definition 6.2. A homogeneous polynomial vector field of degree 2 (a quadratic vector field +for short) on C3 is Halphen if it satisfies the following relation +(10) +[C, X] = 2R , +where C stands for a constant vector field and R is the Radial vector field. +The normal form indicated in (9) is obtained as the solutions of Equation (10) for C = +∂/∂z1 + ∂/∂z2 + ∂/∂z3. Since both C and X are homogeneous, Euler relations imply that we +also have [R, C] = −C and [R, X] = X. In turn, these three relations together mean that the +triplet {R, C, X} generates the Lie algebra of SL (2, C). +Let FX, FR, and FC denote the 1-dimensional foliations associated to the vector fields X, +R and C, respectively. Once again, let �C3 denote the blow-up of C3 centered at the origin with +projection π : �C3 → C3. The exceptional divisor π−1(0) is isomorphic to CP2 and the blow ups +of FX, FR, and FC will respectively be denoted by �FX, �FR, and �FC. Similarly, � +X, �R, and �C will +stand for the blow ups of X, R, and C. Next, recall that, whenever two vector fields commute, +then the flow of one of them will preserve the foliation associated with the other. This simple +remark hints at a basic property of Halphen vector fields. Indeed, since X and C commute up to + +GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS +33 +the Radial vector field, the flow of X “tends” to preserve the projection of the foliation arising +from C along the orbits of R. To make this remark accurate, we first note that the space of +orbits of R is naturally identified with the exceptional divisor π−1(0) ≃ CP2 though, on π−1(0), +� +X vanishes identically and �C has poles. However, the restrictions �FX|π−1(0) and �FC|π−1(0) to +π−1(0) of the foliations �FX and �FC have a specific property of “mutual transversality” which +is reminiscent from the previous observation on commuting vector fields. This can be stated as +follows: +Definition 6.3. Two (singular) foliations F1 and F2 are said to be mutually transverse if they +are (regular and) transverse away from an algebraic curve C which, in addition, is invariant by +both F1 and F2. In particular, the curve C contains all singular points of F1 and of F2. +Keeping in mind that �FC|π−1(0) is nothing but a pencil of projective lines, the “mutual +transversality” condition makes it easy to work out the structure of �FX|π−1(0) directly on +π−1(0) ≃ CP2. Namely, we have: +• Generically, �FX|π−1(0) leaves exactly 3 projective lines C1, C2 and C3 invariant. These +projective lines belong to the pencil �FC|π−1(0) and they intersect mutually at a radial +singularity in π−1(0) (the base locus of the pencil) which is given in homogeneous +coordinates by [1, 1, 1]. Also, the eigenvalues of �FX|π−1(0) at [1, 1, 1] are 1 and 1 (radial +singularity). +• In fact, [1, 1, 1] is a radial singularity for the foliation in the 3-dimensional space. In +other words, the eigenvalue of �FX at [1, 1, 1] associated to the direction transverse to +the exceptional divisor is again 1. +• Away from the invariant projective lines C1, C2, and C3, the foliation �FX|π−1(0) is +transverse to the remaining projective lines in the pencil �FC|π−1(0). +Next, since X is homogeneous, the dynamics of the foliation �FX on �C3 can basically be recovered +from the dynamics of the core foliation �FX|π−1(0) on π−1(0) ≃ CP2. We will return to this point +later. +In view of the preceding, let us first focus on the core foliation �FX|π−1(0). Note that both +�FX|π−1(0) and the pencil �FC|π−1(0), the latter viewed as foliation, share the singular point +[1, 1, 1] ∈ π−1(0). +Consider the (2-dimensional) blow up of π−1(0) ≃ CP2 at [1, 1, 1]. +The +resulting surface is the Hirzebruch surface F1, the CP1-bundle over CP1 with a section of +self-intersection −1. +On F1, the foliation (pencil) �FC|π−1(0) becomes the standard fibration +P : F1 → CP1. In turn, the transform FX,F1 of the foliation �FX|π−1(0) on F1 is regular on a +neighborhood of the −1-rational curve of F1 (identified with the exceptional divisor π−1(0) of +the blow up of CP2). Also, there are 3 fibers of P that are invariant by FX,F1 and these fibers +will still be denoted by C1, C2, and C3 by evident reasons. Away from these three fibers, FX,F1 +is regular and transverse to the fibration induced by P on the open manifold F1 \ {C1, C2, C3}. +The dynamics of � +FX|π−1(0) can naturally be read off the dynamics of FX,F1 which, in turn, +is essentially described by the holonomy representation. +In fact, the restriction of FX,F1 to +the open surface (F1 \ {C1, C2, C3}) is transverse to the restriction to (F1 \ {C1, C2, C3}) of +the fibration P : F1 → CP1. +Since the fibers of P are compact, Ehresmann’s observation +ensures that the restriction of P to the leaves of FX,F1 yields a covering map from the leaf in +question to CP1 \ {z1, z2, z3}, where z1, z2, z3 are in natural correspondence with the invariant +fibers C1, C2, C3. +The dynamics of FX,F1 is therefore essentially encoded in the holonomy +representation, namely: the homomorphism ρ from the fundamental group of CP1 \ {z1, z2, z3} + +34 +J. REBELO AND H. REIS +to the group of automorphisms of the fiber of P arising from parallel transport along leaves of +FX,F1. +Let π1(CP1 \ {z1, z2, z3}) denote the fundamental group of CP1 \ {z1, z2, z3}. Since FX,F1 is +holomorphic, the image of the holonomy representation ρ is contained in the group of holomor- +phic diffeomorphisms of CP1 which can be identified with PSL (2, C). The holonomy group +Γ of FX,F1 is the image of π1(CP1 \ {z1, z2, z3}) by ρ, i.e. +Γ ⊂ PSL (2, C) is defined by +Γ = ρ[π1(CP1 \ {z1, z2, z3})]. +Next, for each i = 1, 2, 3, let ξi ∈ PSL (2, C) be the holonomy map obtained by lifting a +small loop around zi ∈ CP1 in the leaves of FX,F1. The M¨oebious transformations ξ1, ξ2, ξ3 +clearly generate the holonomy group Γ and satisfy the relation ξ1 ξ2 ξ3 = id. With the evident +identifications, the dynamics of Γ on CP1 also accounts for the global dynamics of the foliation +�FX|π−1(0) on CP2. +All of the preceding considerations apply to every Halphen vector field in the family defined +by (9), regardless of whether or not they are semicomplete. To detect semicomplete Halphen +vector fields in the family (9), we proceed as follows. First, notice that the singularities of �FX +and of �FX|π−1(0) do coincide. Naturally there is the point [1, 1, 1] lying at the intersection of +all the lines in the pencil �FC|π−1(0). Moreover, around [1, 1, 1], the foliation �FX is conjugate to +the radial vector field in dimension 3. +To describe the structure of the remaining singular points, for i = 1, 2, 3, let mi = (α1 +α2 + +α3 − 2)/αi provided that αi ̸= 0, and set mi = ∞ otherwise. The remaining singular points of +�FX are contained in the lines C1, C2, C3 and are as follows. +(1) If mi ̸= ∞. Then, aside from [1, 1, 1], �FX possesses exactly two singular points pi and qi +in the line Ci. The eigenvalues of �FX at pi are −1, −1, −mi while at qi the eigenvalues +are −1, −1, mi. In both cases, the eigenvalues are ordered so that to the first eigenvalue +corresponds to a direction transverse to the exceptional divisor, the second eigenvalue is +associated with the direction of Ci and the third eigenvalue is associated with a direction +transverse to Ci and contained in the exceptional divisor. +(2) If mi = ∞. Then, aside from [1, 1, 1], �FX possesses a unique singular point pi = qi in +Ci. At this singular point, the eigenvalues are −1, 0, −1 with the same ordering used in +the above item. +When mi = ∞, the holonomy map ξi is a parabolic map in PSL (2, C) since �FX|π−1(0) has a +(2-dimensional) saddle-node singularity at pi = qi with strong invariant manifold transverse to +Ci. Next, we have: +Proposition 6.4. Assume that X is semicomplete and that mi ̸= ∞. Then mi is an integer +(which can be assumed positive up to reversing the roles of pi and qi). Moreover the holonomy +map ξi : CP(1) → CP(1) is periodic of period mi. +Proof. Again, up to renaming pi and qi, the singular point qi of �FX lies in the Siegel domain and +the eigenvalues of the mentioned foliation at the singular point in question fulfill the conditions +1., 2., 3. and 4. of Theorem 1 in [50] (or, equivalently, Theorem 2.19 in [43]). Furthermore, +with the language of [50], [43], the eigenvalue that can be “separated” from the others by a +straight line through 0 ∈ C is the first eigenvalue (i.e. the eigenvalue associated with direction +transverse to the exceptional divisor). Consider then the separatrix S of �FX tangent to this +direction. It is immediate to check that the restriction of � +X to S is given, in local coordinates, +by −z2∂/∂z. Being X semicomplete, there follows that the local holonomy map of �F arising +from a small loop in S encircling qi must agree with the identity, c.f. [13]. Theorem 1 in [50] +then ensures that � +FX is linearizable around qi. + +GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS +35 +In particular, the foliation �FX|π−1(0) is also linearizable around qi. It follows that �FX|π−1(0) +possesses a separatrix transverse to Ci and that the holonomy map arising from this separatrix is +locally conjugate to a rotation of angle 2π/mi. Because �FX|π−1(0) is transverse to a fibration, this +local holonomy map is, in fact, the restriction of a global M¨oebius transformation ξi ∈ PSL (2, C) +which, therefore, must verify ξmi +i += id. +□ +As an immediate consequence, we have the following +Proposition 6.5. If X is semicomplete, then the holonomy group Γ ⊂ PSL (2, C) describing +the global dynamics of �FX|π−1(0) is given by +Γ =< ξ1, ξ2, ξ3 : ξm1 +1 += ξm2 +2 += ξm3 +3 += ξ1ξ2ξ3 = id > . +In other words, Γ is a triangular group. +In Proposition 6.5, when mi = ∞, the condition ξ∞ +i += id must be interpreted as simply +saying that ξi is parabolic. +In the sequel, we shall also use the convention that 1/mi = 0 +provided that mi = ∞. In order to obtain semicomplete Halphen vector fields with complicate +dynamics, we assume also that +(11) +m = 1 +m1 ++ 1 +m2 ++ 1 +m3 +< 1 . +The effect of inequality (11) is just to rule out finitely many cases where the group Γ is +“elementary”, either finite or conjugate to a subgroup of the affine group of C. +Assuming +m1, m2, m3 fixed and as in (11), the resulting triangular group Γ satisfy all of the following +conditions: +• The group Γ is unique (up to conjugation). +• Γ is discrete and non-elementary. +• Γ leaves a real projective line in CP1 invariant so that Γ is actually a non-elementary +Fuchsian group (i.e. Γ can also be viewed as a subgroup of PSL (2, R)). +• The limit set Λ(Γ) of Γ coincides with the invariant circle S1. In particular, Γ acts +properly discontinuously on each connected component of CP1 \ Λ(Γ). +As is well known, the dynamics of Γ on its limit set Λ(Γ) = S1 is very non-trivial: the +dynamics has all orbits are dense and it is ergodic with respect to the Lebesgue measure. Also +stationary measures are unique (and hard to understand in detail). Clearly, these issues are +directly reflected in the saturated of Λ(Γ) by the foliation �FX|π−1(0) whose dynamics is hence +very non-trivial as well. +It is also convenient to say a few words on the actual dynamics of �FX on �C3 rather than +limiting ourselves to its core foliation �FX|π−1(0). +To describe this dynamics, we can follow +essentially the same ideas used to describe the foliation �FX|π−1(0). Beginning with the pencil +�FC|π−1(0), we define a family of surfaces in �C3 by considering the preimage of each line in +�FC|π−1(0) by the canonical projection Π : �C3 → π−1(0) ≃ CP2. +More precisely, for every +projective line D in the pencil �FC|π−1(0), Π−1(D) is the line bundle over CP1 whose Chern +class equals −1. +Alternatively, by adding a “section at infinity”, Π−1(D) can naturally be +compactified into the Hirzebruch surface F1. In other words, up to adding a “plane at infinity” +to �C3, we obtain a family of F1 surfaces parameterized by the lines in the pencil �FC|π−1(0). Now, +if we remove the three Hirzebruch surfaces sitting on the top of the lines in �FC|π−1(0) that are +invariant under �FX|π−1(0), it is straightforward to conclude that �FX is transverse to the fibration +by F1-surfaces over CP1 \ {z1, z2, z3}. Thus, once again we obtain a representation ρ from the + +36 +J. REBELO AND H. REIS +fundamental group of CP1 \ {z1, z2, z3} in the group Aut (F1) of holomorphic diffeomorphisms +of F1. Let Γ be the image of ρ, i.e. the holonomy group of �FX. Clearly, ρ is generated by the +maps Ξi obtained by lifting a small loop around zi, i = 1, 2, 3. The maps Ξi can explicitly be +computed. Fix a surface F1 equipped with coordinates (x, w) where x is projective coordinate +on the projective line Π(F1) and w is an affine coordinate on the fibers of F1 that equals zero +in the intersection with the exceptional divisor. Then we have +(12) +Ξi(x, w) = (ξi(x), +� +ξ′(x) w) +c.f. [45]. Keeping in mind that the dynamics of �FX on �C3 and the dynamics of FX on C3 can +be identified, what precedes can be summarized as follows (the slight abuse of language should +not really lead to any misunderstanding): +Proposition 6.6. The dynamics of FX on C3 is essentially equivalent to the dynamics of the +group Γ = ⟨Ξ1, Ξ2, Ξ3⟩ on F1. In particular, the (−1)-section of F1 is invariant by Γ and the +restriction of the action of Γ to this section is nothing but the action of the triangular group Γ +on CP1. +By now, we have provided a description of the (rather non-trivial) dynamics of Halphen +vector fields such that the quantities mi are integers satisfying the condition in (11) and the +reader is referred to [17] for additional information. However, strictly speaking, we still do not +know whether or not Halphen vector fields satisfying the conditions in question are, indeed, +semicomplete. In fact, Proposition 6.4 provides only necessary conditions for the vector field to +be semicomplete. Hence, there remains the problem of checking that these conditions are also +sufficient. +Curiously enough the fact that the corresponding Halphen vector fields are semicomplete is +basically included in Halphen original papers [20], [21]. Halphen begins his Note by pointing +out that, if φ is a solution of a Halphen vector field, then so is +(13) +�φ = +1 +(ct + d)2 φ +�at + b +ct + d +� +− +c +ct + d , +for every a, b, c, d ∈ C with ad − bc ̸= 0. From this he concludes that all solutions can be +described out of a single “known” solution. He then goes on to obtain a particular solutions +by skillfully manipulating theta functions. In this sense, the converse to Proposition 6.4 can be +derived from his work. +Yet, Guillot [17] provides a different proof of the semicomplete nature of Halphen vector +fields satisfying the conditions in in Proposition 6.4. Guillot’s argument dispenses with the +remarkable identities satisfied by theta functions and, perhaps more importantly, lends itself +well to deep generalizations. We will close this paragraph by sketching this argument. +First, it is convenient to recall the basic notions of translation, affine, and projective structures +on Riemann surfaces since they play a role in the discussion below. +Definition 6.7. Let S be a Riemann surface along with a covering {(Bi, ϕi)} by local coordi- +nates. The covering {(Bi, ϕi)} is said to define a translation structure (resp. affine structure, +projective structure) on S if and only if the changes of coordinates ϕi ◦ ϕ−1 +j +: ϕj(Bi ∩ Bj) → +ϕi(Bi ∩ Bj) are restrictions of translations of C (resp. affine maps, M¨oebius transformations). +In particular, if S is endowed with a nowhere zero holomorphic vector field X, then the +covering whose local coordinates are the inverse maps of the (local) solutions of X endows S +with a translation structure. This simple remark will be useful below. +Also, a translation structure (resp. +affine structure, projective structure) gives rise to a +monodromy homomorphism ρ from the fundamental group of S to the group of translations of + +GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS +37 +C (resp. affine maps, M¨oebius transformations). Following [17], [19], denote by Sρ the covering +space of S associated with the kernel of ρ. On Sρ, we can define a developing map Dρ : Sρ → C +(resp. C, CP1). In fact, Sρ is the smallest covering of S on which a developing map is well +defined. This developing map will be called the monodromy-developing map of the corresponding +structure. Naturally, all developing maps are well defined up to composition with an element +of the corresponding group (translation, affine map, or M¨oebius transformations). +Remark 6.8. The preceding offers us yet another equivalent way to define semicomplete vector +fields on a Riemann surface, and thus in general since a vector field will be semicomplete if and +only if its restriction to each leaf of its associated foliation is semicomplete. Namely, the vector +field X on the Riemann surface S is semicomplete if and only if the monodromy-developing map +of the corresponding translation structure is injective. We are now ready to explain Guillot’s +argument. +Sketch of Guillot’s proof that Halphen vector fields as in Proposition 6.4 are semicomplete. We +might start by recalling that the vector fields R and C generate the Lie algebra of the affine group +Aff (C, 0). Of course a similar remark applies to their blow ups �R and �C. Then we consider +the Zariski open subset W of �C3 given as the complement of π−1(0) and of the 3 invariant +Hirzebruch surfaces. In the setting of Proposition 6.6, W is a U-bundle over CP1 \ {z1, z2, z3}, +where U ⊂ F1 is the Zariski open set defined as the complement of the two rational sections of +F1. In particular, U is in a natural correspondence with an orbit of Aff (C, 0). Finally, �FX is +transverse to the fibers of the fibration W → CP1 \ {z1, z2, z3} and admits a global holonomy +group determined by Proposition 6.6. +It suffices to show that the restriction of � +X to W is semicomplete. Guillot basic observation +is that � +X induces a natural projective structure on CP1 \ {z1, z2, z3} viewed as the base of the +U-bundle W. This deserves a few comments. Small discs B ⊂ CP1\{z1, z2, z3} can be identified +with discs on the leaves L of �FX by means of the fiber bundle structure. Next, each leaf L +of the restriction of �FX to W is endowed with a translation structure induced by � +X. These +transverse structures vary with the leaf but its underlining projective structure does not. In +fact, taking into account that a fiber U of the U-bundle W is identified with an orbit of Aff (C) +- and thus parameterized by the flows of R and of C, Equation (13) can be interpreted as an +identity involving the flows of R, C, and X (or of their blow ups which amounts to the same). +With this interpretation, it becomes clear that the time taken by � +X to move between two fixed +fibers U1 and U2 along leaves L and L′ are related by a M¨oebius transformation. Thus the +covering of the base CP1 \ {z1, z2, z3} obtained by taking the inverses of the local solutions of +X, as above, over all possible leaves of � +FX defines a projective structure on CP1 \ {z1, z2, z3}. +Next, consider the monodromy-developing map Dρ for the projective structure CP1\{z1, z2, z3}. +It is straightforward to check that representatives for this developing-map can be obtained by +simply considering the monodromy-developing maps associated with the translation structures +induced by � +X on the leaves of �FX (or more accurately of the restriction of �FX to W). In view +of Remark 6.8, there follows that X is semicomplete if and only if Dρ is injective. Guillot’s then +“compute” the projective structure in question by means of the Schwarzian operator so as to +show that Dρ is essentially Schwarz triangular functions and the proof follows. +□ +Remark 6.9. In fairness, we should note that the material covered in this paragraph is essentially +the first part of Guillot’s paper [17]. The content of [17] also includes realizing semicomplete +Halphen vector fields as actual complete vector fields on complex manifolds as well as several +important applications to the study of SL (2C) actions and homogeneous spaces. + +38 +J. REBELO AND H. REIS +Let us close this paragraph with a couple of questions about dynamics of semicomplete vector +fields, the first one being kind of inevitable. +Problem 1. Are there semicomplete vector fields with complicated dynamics which gen- +uinely different from the dynamics obtained by means of Halphen vector fields? +Another interesting question which may or may not have a saying in the above problem +concerns geodesic flows on semisimple Lie groups. +These geodesic flows have already been +considered in works by S. Dumitrescu and by Elshafei-Ferreira-Reis, see [10], [12] and their +references. Given a (semisimple) Lie group G and a left-invariant holomorphic metric on ⟨ . ⟩ +on G, the complex geodesic flow on G can be expressed by a quadratic vector field defined on +the Lie algebra of G by means of the Euler-Arnold formalism. This yields a particular, yet +large and with geometric nature, class of quadratic vector fields. Referring to vector fields in +this class as Euler-Arnold vector fields, their dynamics is definitely worth study. Thus we can +formulate the following special case of the preceding question which, however, holds interest in +its own: +Problem 2. Are there semicomplete Euler-Arnold vector fields exhibiting complicated dy- +namical behavior? +6.2. Local aspects of semicomplete vector fields and applications. Partly, the interest +of semicomplete vector fields comes from the fact that they provide local obstructions for a +germ of vector field be realized as singularity of a complete one. In the sequel, we will talk +about germs of semicomplete vector fields or about semicomplete singularities as synonymous. +From the basic properties discussed at the beginning of this section, it follows that semicom- +plete vector fields can be viewed as a “local counterpart” of complete ones. In fact, a singularity +that is not semicomplete cannot be realized by a complete vector field. In particular, it cannot +be realized by a globally defined holomorphic vector field on a compact manifold. The un- +derstanding of semicomplete singularities is therefore useful to the description of holomorphic +vector fields (globally) defined on compact manifolds. +To better explain this issue, it is convenient to center the discussion around a rather concrete +and well known question due to E. Ghys that can be formulated in terms of semicomplete +vector fields as follows: let X be a semicomplete holomorphic vector field on (Cn, 0) with isolated +singular points. Is it true that J2X(0) ̸= 0, i.e. must the second jet of X at the singular point +be different from zero ? +Ghys’ original motivation seems to be related to problems about bounds for the dimension of +automorphism group of compact complex manifolds. To be more precise, consider a compact +complex manifold M and denote by Aut (M) the group of holomorphic diffeomorphisms of M. +It is well known that Aut (M) is a finite dimensional complex Lie group whose Lie algebra can +be identified with X (M), the space of all holomorphic vector fields defined on M. A too na¨ıve +question, would be to wonder if the dimension of Aut (M) can be bounded by a function of +the dimension of M. It turns out, however, that the dimension of the automorphism group +of the Hirzebruch surface Fn is n + 5 provided that n ≥ 1. In particular, already in the case +of compact surfaces, the dimension of Aut (M) can be arbitrarily large. However, analogous +questions can be raised to better effect for specific classes of manifolds. For example, among +projective manifolds with Picard group isomorphic to Z, Hwuang and Mok asked if there is a +n-dimensional manifold whose dimension of the automorphism group exceeds the dimension of +the automorphism group of CPn. +As a matter of fact, Ghys’ question is part of a general principle with vaguely stated as follows: +semicomplete singularities cannot be “too degenerate”. Here it is convenient to explain how +limiting the extent to which a semicomplete singularity can be degenerate becomes a useful tool + +GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS +39 +to deal with the previous questions. Consider a n-dimensional compact complex manifold M +and let Aut (M) and X (M) be as above. Fix a point p ∈ M and let k ∈ N be given. Finally, +let Xk +p(M) stand for the set of holomorphic vector fields with vanishing k-jet at p and denote +by Jk +p (M) the space of k-jets at p. The natural mappings +Xk +p (M) → X (M) → Jk +p (M) , +give rise to a short exact sequence so that we have +dim X (M) ≤ dim Xk +p (M) + dim Jk +p (M) . +The dimensions of the jet spaces Jk +p (M) are explicitly given in terms of k and of n = dim (M). +In particular, if for some p ∈ M and k ∈ N, we can obtain bounds for dim Xk +p (M) in terms +of dim (M) then bounds for dim (Aut (M)) follow immediately. For example, suppose that we +happen to know that for a certain class of compact manifolds every singularity of a globally +defined holomorphic vector field is necessarily isolated. Then, assuming Ghys conjecture holds, +it follows that dim X3 +p (M) = 0 and therefore the dimension of Aut (M) would be bounded by +(n3 + 3n2 + 2n)/2. Of course, in general, non-isolated singularities also appear so that it is +convenient to be able to handle them as well. +Aside from introducing the notion of semicomplete singularity, the content of [41] can fairly +be summarized by the following theorem: +Theorem 6.10. [41] Let X be a holomorphic semicomplete vector field on (C2, 0). If the origin +is an isolated singular point for X, then J2 +0X ̸= 0. +The proof of Theorem 6.10 relies on Camacho-Sad theorem on the existence of separatrices +for foliations on (C2, 0). Indeed, since the singular set of X is reduced to the origin, the restric- +tion of X to any analytic invariant curve going through the origin cannot vanish identically. +Furthermore, this restriction is still a semicomplete vector field. Considering then the restric- +tion of X to a separatrix, whose existence is ensured by Camacho-Sad theorem, the problem +becomes essentially reduced to the one-dimensional situation (whether or not the separatrix +is smooth). The resulting (one-dimensional) problem is settled in the same paper by direct +methods. +The question on whether or not Ghys conjecture holds for semicomplete vector fields in higher +dimensions is hence natural. The first deep investigations involving semicomplete vector fields +in higher dimensions were conducted by A. Guillot in [16], and [17] (here “higher” means ≥ 3). +The mentioned papers by Guillot contain, in particular, numerous examples of quadratic semi- +complete vector fields exhibiting a wide range of geometric and dynamical behaviors. Among +these examples, we have already discussed the case of Halphen vector fields that have compli- +cated dynamics and no (non-trivial) holomorphic/meromorphic first integral (cf. Section 6.1). +Moreover, Guillot’s work also make clear that in dimensions ≥ 3, an exhaustive classification +of all semicomplete vector fields with zero linear part - paralleling the list provided in [13] - is +unlikely to exist or, at least, it would be too long to be truly useful. +This is therefore a good moment to elaborate on the difficulties in extending to (C3, 0) +the general classification results in dimension 2 of [41], [13], not to mention the more general +results of [19] encompassing also meromorphic vector fields. Indeed, whether or not obtaining +these generalizations is a tall order, it certainly seems useful to explicitly list some of the new +difficulties arising in dimensions greater than 2. Aside from the existence of core dynamics, +that is a general difficulty already emphasized in this work, the following issues are worth +mentioning. + +40 +J. REBELO AND H. REIS +1. The basic approach to Ghys conjecture stemming from [41] consists of finding a sep- +aratrix. Namely, the following holds: let X be a semicomplete (holomorphic) vector +field on (Cn, 0) with an isolated singularity at the origin. If X possesses a separatrix, +then J2 +0 X ̸= 0. However, as previously seen, Gomez-Mont and Luengo [15] have proved +that separatrices do not exist in general for germs of 1-dimensional foliations on (Cn, 0), +n ≥ 3 (cf. Section 5). +2. The examples provided in [15], however, are not semicomplete so that it is conceivable +that all semicomplete vector field possesses a separatrix. While this seems to suggest +that Ghys conjecture may be proved by showing that semicomplete vector fields do have +separatrices, a direct approach to the latter question does not seem feasible. +3. A more promising point of view regarding item 2 above consists of noticing that the +detailed classification of semicomplete vector fields in dimension 2, as developed in [13] +or in [19], dispenses with Camacho-Sad theorem. In fact, these deeper analysis yield +directly the classification. Hence the existence of separatrices for semicomplete vector +fields on (C2, 0) becomes a corollary, as opposed to a statement needed a priori. +4. In dimension 2, a fundamental ingredient permeating virtually all works on singularities +of vector fields or foliations is the resolution theorem of Seidenberg [55]. Since resolu- +tions theorems for 1-dimensional foliations have been established in the past few years, +c.f. Section 4, this initial difficulty has now been overcome. In fact, as far as semicom- +plete vector fields are concerned, a totally faithful analogue of Seidenberg’s theorem is +available in dimension 3 as will be seen below. +5. Difficulties, however, are not limited to reduction of singularities procedures nor to the +phenomenon of core dynamics. For example, assume our objective is to establish Ghys +conjecture by means of proving the existence of separatrices (in which case the role +played by core dynamics is significantly reduced, c.f. Section 5.2). Assume, in addition, +that we are given a holomorphic foliation admitting a simple reduction of singularities. +In dimension 3, the existence of saddle-node singularities appearing in the resolution +procedure cannot easily be ruled out. In particular, codimension 2 saddle-nodes (i.e. +with two eigenvalues equal to zero) may appear and these singularities are still poorly +understood. +The remainder of this paper is to complement the above list with further comments and +results, some of them proposing simpler approaches that can be effective pending on the specific +application targeted. +Concerning items 1 and 2, some partial results have been proved in [47]. In fact, recall that +Theorem 5.8 states that in the case we are given two holomorphic vector fields X and Y yielding +a representation of a Lie algebra of dimension 2 and not everywhere parallel, then they possess +a common separatrix. By elaborating on this theorem, the following weaker version of Ghys +conjecture in dimension 3 was proven in [47]: +Theorem 6.11. [47] Consider a compact complex manifold M of dimension 3 and assume that +the dimension of Aut (M) is at least 2. Let Z be an element of X (M) and suppose that p ∈ M +is an isolated singularity of Z. Then +J2(Z) (p) ̸= 0 , +i.e. the second jet of Z at the point p is different from zero. +The reader will note that, as far as estimates on the dimension of automorphism groups +are concerned, Theorem 6.11 is as effective as an affirmative solution to Ghys conjecture in + +GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS +41 +dimension 3 in the sense that if the additional assumption needed for Theorem 6.11 is not +verified, then the dimension of Aut (M) is at most 1. +One of the advantages of looking for bounds for the dimension of Aut (M) by means of local +considerations is that the results obtained are essentially valid for open manifolds as well. For +example, studying finite dimensional Lie group actions on Stein manifolds is an active topic in +several complex variables whose roots lie in a classical work of Suzuki [56]. In this direction, +our techniques yield: +Theorem 6.12. [47] Let N denote a Stein manifold of dimension 3 and consider a finite +dimensional Lie algebra G embedded in Xcomp(N) (the space of complete holomorphic vector +fields on N). Assume that the dimension of G is at least 2. If Z is an element of G ⊆ X (M) +possessing an isolated singular point p ∈ N, then the linear part of Z at p cannot vanish, i.e. p +is a non-degenerate singularity of Z. +We may point out that the automorphism group of a Stein manifold is not a finite dimensional +Lie group in general. Indeed, even C2 has an infinite dimensional group of automorphisms with +hardly any non-trivial structure of Lie group. This difficulty is avoided in the statement of +Theorem 6.12 by the assumption that, from the beginning, we are dealing with some finite +dimensional Lie algebra: owing to Lie theorem, such Lie algebra can always be integrated to +yield a (complete) action of the corresponding Lie group. This part of the statement actually +holds for arbitrary complex manifolds of dimension 3 (whether or not they are compact or +Stein) and parallels Theorem 6.11 in the sense that the second jet of a vector field Z ∈ G will +never be zero at an isolated singular point. The role played by the Stein condition is to ensure +that the first jet, rather than the second one, is different from zero at isolated singular points. +To close this paper, let us go back to resolution theorems as discussed in Section 4 and further +sharpen the results under the additional assumption of having semicomplete vector fields. As +already mentioned, resolution theorems always play a central role in singularity theory and +sharp resolution statements exist for 1-dimensional foliations in dimension 3. Yet, given their +importance, it is convenient to have available the simplest possible resolution statements in +every circumstance. +In particular, it is natural to wonder if semicomplete singularities or +other special classes of singular points allow for simpler resolution theorems facilitating a more +detailed analysis of their structures. +The possibility of having simpler resolution theorems valid for semicomplete singularities was +also considered in [51] whose initial motivation was, in fact, to obtain a resolution theorem for +semicomplete singularities that would faithfully parallel Seidenberg theorem for foliations on +(C2, 0). This type of statement is useful to approach problems such as Ghys conjecture or to +investigate compact complex manifolds of dimension 3 equipped with holomorphic vector fields. +In this setting, Theorem 6.13 below is proved in [51]. +Theorem 6.13. [51] Let X be a semicomplete vector field defined on a neighborhood of the +origin of C3 and denote by F the holomorphic foliation associated with X. Then one of the +following holds: +(1) The linear part of X at the origin is nilpotent (non-zero). +(2) There exists a finite sequence of (standard) blow-ups along with transformed foliations +F = F0 +π1 +←− F1 +π2 +←− · · · +πr +←− Fr +such that all of the singular points of Fr are elementary. Moreover, each blow-up map +πi is centered in the singular set of the corresponding foliation Fi−1. In other words, +the foliation F can be resolved by means of standard blow-ups. + +42 +J. REBELO AND H. REIS +Let us emphasize that item 1 in Theorem 6.13 involves the linear part of the vector field +X rather than the linear part of the associated foliation F. In fact, it is the linear part of X +that has to be (nilpotent) non-zero from the outset. Furthermore, this property is “universal” +in the sense that it does not depend on any sequence of blow-ups/blow-downs carried out. In +particular, we can choose a “minimal model” for our manifold and the corresponding transform +of X will still have non-zero nilpotent linear part at the corresponding point. +From what +precedes, the following also deserves to be singled out. +Corollary 6.14. [51] Let X be a semicomplete vector field defined on a neighborhood of +(0, 0, 0) ∈ C3 and assume that the linear part of X at the origin is equal to zero. Then item (2) +of Theorem 6.13 holds. +Accurate normal forms for persistent nilpotent singular points were provided in the same +paper, c.f. Theorem 4.4. However, not all of them need to be semicomplete or, indeed, realized +as singularity of a complete flow. Taking into account the global setting of complete vector +fields, it is natural to wonder if there is, indeed, complete vector fields inducing a foliation with +singular points that cannot be resolved by standard blow ups as in item (2) of Theorem 6.13. +As a matter of fact, these singularities do exist and an explicit example is provided by the +polynomial vector field +Z = x2 ∂ +∂x + xz ∂ +∂y + (y − xz) ∂ +∂z . +Although Z is not complete on C3, it can be extended to a complete vector field defined on +a suitable open manifold. In particular, the point corresponding to the origin of the above +coordinates (x, y, z) constitutes a nilpotent singular point of Z that cannot be resolved by +means of standard blow-ups with centers in the singular set. +Finally, we might emphasize that the example above involves a complete vector field defined +on an open manifold. We might then ask if this phenomenon still occurs in the far more restric- +tive context of compact manifolds of dimension 3. Since in the compact case the completeness +condition becomes automatic, we are simply asking whether or not there is a compact complex +manifold of dimension 3 equipped with a (global) holomorphic vector field X which exhibits a +singular point that cannot be resolved by means of standard blow ups. This time, the answer +turns out to be negative as the following holds: +Corollary 6.15. Let F be the foliation associated with a vector field X globally defined on some +compact complex manifold M of dimension 3. Then every singular point of F can be resolved +by a sequence of standard blow ups. +In closing, let us just point out that both Corollary 6.14 and Corollary 6.15 are strictly speak- +ing by-products of the methods used to prove Theorem 6.13 rather than formal consequences +of the statement of this theorem. +References +[1] W. Barth, K. Hulek, C. Peters, & A. Van de Ven, Compact complex surfaces, second edition, Springer- +Verlag, Berlin, (2004). +[2] C.A. Briot & J.C. Bouquet, Propri´et´es des fonctions d´efinies par des ´equations diff´erentielles, Journal +de l’Ecole Polytechnique, 36 (1856), 133-198. +[3] C. Camacho, Problems on Limit Sets of Foliations of Complex Projective Spaces, International Congress +of Mathematicians (Kyoto), Springer Verlag, (1990), 1235-1239. +[4] C. Camacho, Quadratic forms and holomorphic foliations on singular surfaces, Matematische Annalen, +282, (1988), 177-184. +[5] C. Camacho & P. Sad, Invariant Varieties through Singularities of Holomorphic Vector Fields, Annals of +Math., 115 (1982), 579-595. + +GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS +43 +[6] F. Cano, Desingularization Strategies for Three Dimensional Vector Fields, Lecture Notes in Mathematics, +1259, Springer-Verlag, Berlin, (1987). +[7] F. Cano & D. Cerveau, Desingularization of non-dicritical holomorphic foliations and existence of sepa- +ratrices, Acta Math., 169, (1992), 1-103. +[8] F. Cano, Reduction of the singularities of codimension one singular foliations in dimension three, Ann. of +Math., 160, 3, (2004), 907-1011. +[9] F. Cano, C. Roche & M. Spivakovsky, Reduction of singularities of three-dimensional line foliations, +Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, February 2013, +DOI: 10.1007/s13398-013-0117-7. +[10] S. Dumitrescu, M´etriques riemanniennes holomorphes en petite dimension, Ann. Inst. Fourier (Grenoble), +51, 6, (2001), 1663-1690. +[11] P. Elizarov, Y. Il’yashenko, A. Shcherbakov, & S. Voronin, Finitely generated groups of germs of +one-dimensional conformal mappings and invariants for complex singular points of analytic foliations of the +complex plane, Adv. in Soviet Math. 14, (1993). +[12] A. Elshafei, A.C. Ferreira & H. Reis, Geodesic completeness of pseudo and holomorphic Riemannian +metrics on Lie groups, preprint available from arXiv: https://arxiv.org/abs/2208.10873. +[13] E. Ghys & J.C. Rebelo, Singularit´es des flots holomorphes II, Ann. Inst. Fourier (Grenoble), 47, 4, (1997), +1117-1174. +[14] P.A. Griffiths & J. Harris, Principles of Algebraic Geometry, Wiley Classics Library. John Wilet & Sons +Inc., New York (1994). +[15] X. Gomez-Mont & I. Luengo, Germs of holomorphic vector fields in C3 without a separatrix, Invent. +Math., 109, 2 (1992), 211-219. +[16] A. Guillot, Semicompleteness of homogeneous quadratic vector fields, Ann. Inst. Fourier (Grenoble), 56, +5, (2006), 1583-1615. +[17] A. Guillot, Sur les ´equations d’Halphen et les actions de SL (2, C), Publ. Math. IHES, 105, 1, (2007), +221-294. +[18] A. Guillot, Meromorphic vector fields with single-valued solutions on complex surfaces, Adv. Math., 354, +(2019), 106742, 41 pages. +[19] A. Guillot & J.C. Rebelo, Semicomplete meromorphic vector fields on complex surfaces, Journal fur die +reine und angewandte Mathematik, 667, (2012), 27-65. +[20] G.-H. Halphen, Sur un syst`eme d’´equations diff´erentielles, Comptes Rendus Hebdomadaires de l’Acad´emie +des Sciences, Vol XCII, 24, (1881), 1101-1102. +[21] G.-H. Halphen, Sur certains syst`emes d’´equations diff´erentielles, Comptes Rendus Hebdomadaires de +l’Acad´emie des Sciences, Vol XCII, 24, (1881), 1404-1406. +[22] M. O. Huddai-Verenov, A property of the solutions of a differential equation (Russian), Mat. Sbornik, 56 +(98), 3, (1962), 301-308. +[23] Y. Il’yashenko & S. Yakovenko, Lectures on analytic differential equations, Graduate Studies in Math- +ematics, Vol. 86, American Mathematical Society, Providence, RI, (2008). +[24] Y. Il’yashenko, The topology of phase portraits of analytic differential equations in the complex projective +plane (Russian), Trudy Sem. Petrovsk, 4, (1978), 83-136. (English), Sel. Math. Sov., 5, 2, (1986), 141-199. +[25] E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, (1944). +[26] J.-P. Jouanolou, ´Equations de Pfaff alg´ebriques, Lect. Notes Math., 708, (1979). +[27] J.-P. Jouanolou, Hypersurfaces solutions d’une ´equation de Pfaff analytique, Math. Ann., 232, (1978), +239-245. +[28] F. Loray & J.C. Rebelo, Minimal, rigid foliations by curves in CPn, J. Eur. Math. Soc., 5, (2003), +147-201. +[29] B. Malgrange, Frobenius avec singularit´es, I: codimension un, Publ. Math. IHES, 46, (1976), 163-173. +[30] D. Marin & J.F. Mattei, Monodromy and topological classification of germs of holomorphic foliations, +Ann. Sci. ´Ec. Norm. Sup´er. (4), 45, 3, (2012), 405-445. +[31] D. Marin & J.F. Mattei, Topology of singular holomorphic foliations along a compact divisor, J. Singul., +9, (2014), 122-150. +[32] D. Marin, J.F. Mattei & E. Salem, Topological moduli space for germs of holomorphic foliations, Int. +Math. Res. Notice IRMN, 23, (2020), 9228-9292. +[33] D. Marin, J.F. Mattei & E. Salem, Topological moduli space for germs of holomorphic foliations II: +universal deformations, preprint, https://arxiv.org/abs/2105.12688 +[34] D. Marin, J.F. Mattei & E. Salem, Topological moduli space for germs of holomorphic foliations III: +complete families, preprint, https://arxiv.org/abs/2201.07479 + +44 +J. REBELO AND H. REIS +[35] J.-F. Mattei & R. Moussu, Holonomie et int´egrales premi`eres, Ann. Sc. E.N.S. S´erie IV, 13, 4, (1980), +469-523. +[36] M. McQuillan & D. Panazzolo, Almost ´etale resolution of foliations, preprint IHES, IHES/M/09/51, +(2009). +[37] M. McQuillan & D. Panazzolo, Almost ´etale resolution of foliations, J. Differential Geometry, 95, +(2013), 279-319. +[38] I. Nakai, Separatrizes for non solvable dynamics on (C, 0), Ann. Inst. Fourier, 44, (1994), 569-599. +[39] D. Panazzolo, Resolution of singularities of real-analytic vector fields in dimension three, Acta Math., 197, +no 2 (2006), 167-289. +[40] O. Piltant, An Axiomatic Version of Zariski’s Patching Theorem, Rev. R. Acad. Cienc. Exactas Nat. Ser. +A Math. RACSAM, 107, 1, (2013), 91-121. +[41] J.C. Rebelo, Singularit´es des flots holomorphes, Ann. Inst. Fourier (Grenoble), 46, 2, (1996), 411-428. +[42] J.C. Rebelo, On transverse rigidity for singular foliations in (C2, 0), Ergod. Th. & Dynam. Sys., 31, 3, +(2011), 935-950. +[43] J.C. Rebelo & H. Reis, Local Theory of Holomorphic Foliations and Vector Fields, Lecture Notes available +from arxiv (arXiv:1101.4309). +[44] J.C. Rebelo & H. Reis, Separatrices for C2-actions on 3-manifolds, Commentarii Mathematici Helvetici, +88, 3, (2013), 677-714. +[45] J.C. Rebelo & H. Reis, Uniformizing complex ODEs and Applications, Rev. Mat. Iberoam., 30, 3, (2014), +799-874. +[46] J.C. Rebelo & H. Reis, A note on integrability and finite orbits for subgroups of Diff (Cn, 0), Bull. Braz. +Math. Soc. (NS), 46, 3, (2015), 469-490. +[47] J.C. Rebelo & H. Reis, 2-dimensional Lie algebras and separatrices for vector fields on (C3, 0), Journal +de Math´ematiques Pures et Appliqu´ees, 105, 2, (2016), 248-264. +[48] J.C. Rebelo & H. Reis, Discrete orbits, recurrence and solvable subgroups of Diff (C2, 0), Journal of +Geometric Analysis, 27, 1, (2017), 1-55. +[49] J.C. Rebelo & H. Reis, On resolution of 1-dimensional foliations on 3-manifolds, Russian Mathematical +Surveys, 76, 2, (2021), 291-355. +[50] H. Reis, Equivalence and semi-completude of foliations, Nonlinear Analysis. Theory, Methods and Applica- +tions, 64, 8, (2006), 1654-1665. +[51] H. Reis, The geometry and dynamics of complex ordinary differential equations, Habilitation, University of +Porto, Portugal (2021). +[52] J. Ribon, Recurrent orbits of subgroups of local complex analytic diffeomorphisms, Mathematische +Zeitschrift, 285, (2017), 519-548. +[53] A.A. Shcherbakov, On the density of an orbit of a pseudogroup of conformal mappings and a generalization +of the Hudai-Verenov theorem, Vestn. Mosk. Univ., 31, Ser. I, (1982), 10-15. +[54] A.A. Shcherbakov, Topological and analytic conjugation of non-commutative groups of conformal map- +pings, Tr. Semin. Petrovsk, 10, (1984), 170-192. +[55] A. Seidenberg, Reduction of singularities of the differential equation Ady=Bdx, American Journal of +Mathematics, 90, (1968), 248-269. +[56] M. Suzuki, Sur les op´erations holomorphes de C et de C∗ sur un espace de Stein, S´eminaire F. Norguet, +Springer LNM, 670, (1975-1976), 58-66. +[57] L. Teyssier, Germes de feuilletages pr´esentables du plan complexe, Bull. Braz. Math. Soc. (NS), 46, 2, +(2015), 275-329. +[58] F. Touzet, Feuilletages holomorphes admettant une mesure transverse invariante, Annales de la Fac. des +Sciences de Toulouse, XXIV, 3, (2015), 523-541. +[59] H, Zoladek, The monodromy group, Mathematics Institute of the Polish Academy of Sciences, Mathemat- +ical Monographs (New Series), 67, Birkh¨auser Verlag, Basel, (2006). +Institut de Math´ematiques de Toulouse ; UMR 5219, Universit´e de Toulouse, 118 Route de +Narbonne, F-31062 Toulouse, France. +Email address: rebelo@math.univ-toulouse.fr +Centro de Matem´atica da Universidade do Porto, Faculdade de Economia da Universidade do +Porto, Portugal. +Email address: hreis@fep.up.pt + diff --git a/BdE5T4oBgHgl3EQfTA_5/content/tmp_files/load_file.txt b/BdE5T4oBgHgl3EQfTA_5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..45e20987049a8d5a1df11ce62270aa9524558e23 --- /dev/null +++ b/BdE5T4oBgHgl3EQfTA_5/content/tmp_files/load_file.txt @@ -0,0 +1,1859 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf,len=1858 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='05534v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='DS] 13 Jan 2023 GLOBAL DYNAMICS AND PERSPECTIVES ON SINGULARITIES OF HOLOMORPHIC FOLIATIONS JULIO REBELO AND HELENA REIS Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In dimensions greater than or equal to 3, the local structure of a singular holo- morphic foliation conceals a globally defined foliation on the projective space of dimension one less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In this paper, we will investigate how the global dynamics of the latter foliation exerts influence on several problems that apparently have a purely local nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In the course of the discussion, a few recent results and open problems in the area will be reviewed as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Basics in the local theory of foliations and the special case of dimension 2 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Splitting the problem: core dynamics and resolution 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Resolution theorems in dimension 3 14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Invariant analytic sets 21 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Semicomplete vector fields, automorphism groups, and separatrices 30 References 42 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Introduction All foliations considered in this work are holomorphic and (possibly) singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Whereas our main object are 1-dimensional holomorphic foliations and holomorphic/meromorphic vector fields, foliations of codimension 1 will also play a role in the discussion especially when the am- bient is of dimension 3, see Section 2 for accurate definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Foliations of dimension 1 defined on some complex manifold M will typically be denoted by F while D will stand for codimen- sion 1 foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The purpose of this paper is to discuss recent results and open problems in the local theory of 1-dimensional foliations when the ambient manifold M has dimension 3 though, occasionally, results and questions in higher dimensions will also be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Foliations defined on complex surfaces, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' complex manifolds of dimension 2, are basically left aside in this paper largely due to the fact that their local theory is much more advanced than their higher dimensional counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Indeed, these singularities are only mentioned in Section 2 and, yet, with the simple purpose of identifying a few issues that make them so special and amenable to very detailed analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In doing so, we will be able to single out one of the most fundamental issues guiding the discussion conducted here: the presence of a global dynamical phenomenon intrinsically attached to germs F of 1-dimensional foliations on Cn provided that n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In slightly vague though more incisive words, the understanding of a germ of 1-dimensional foliation F on Cn, n ≥ 3, passes through the description of a foliation defined on CPn−1 which, as a global object, may exhibit a wild dynamical behavior (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The global foliation in question will usually be referred to as the core foliation of the (local) 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Primary 34M35, 32M25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Secondary 34M45, 32M05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 1 2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REBELO AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REIS foliation F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' We will also use the phrase core dynamics of F to refer to the dynamics of the core foliation associated with F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The common thread of this paper is the existence and implications of a global dynamical system inherently attached to the structure of a singularity of a 1-dimensional, holomorphic foliation defined on (Cn, 0) provided that n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Basically, we will discuss which types of results can be proved if the above mentioned dynamics can accurately be described and, similarly, which general problems may provide us with the tools to ensure this dynamics is “tame enough” to be described, while bearing in mind that in full generality this dynamics can be extremely wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In Section 2, we introduce standard terminology and recall some basic features of singular foliations, in particular pointing out fundamental issues setting apart foliations of dimension 1 and foliations of codimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Then we focus on the special case of singularities of foliations defined on (C2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Whereas this case is clearly distinguished by the fact that the foliations are simultaneously of dimension 1 and of codimension 1, we discuss to a rather non-trivial extent the main 2-dimensional issues allowing for the existence of such a sophisticated theory covering truly fine issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In Section 3, we introduce a fundamental object that exists for singularities of (1-dimensional) foliations defined on (Cn, 0) provided that n ≥ 3, namely: the core dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This is a global foliation defined on CPn−1 whose (global) dynamics poses a fundamental obstacle towards the full understanding of the initial singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, we explain how this core dynamics plays a major role in problems about existence of separatrices for codimension 1 foliations on (C3, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Also, we show how its very existence basically rules out any hope of achieving a full understanding of large classes of singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The remainder of this survey is devoted to more advanced material, in particular touching on quite a few open problems of current interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Section 4 contains a detailed review of resolution theorems for singular points of 1-dimensional foliations on (C3, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The first definitive resolution theorem in this context was obtained by McQuillan and Panazzolo in [37] which, in turn, relies heavily on Panazzolo’s algorithm introduced in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' More recently, a different proof based on Zariski general point of view was obtained in [49] which can be seen as the completion of the previous work carried out by Cano-Roche-Spivakovsky [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Despite the undisputed importance of resolution theorem, it seems these results are still not as widely known as one would expect and, for this reason, we thought useful to conduct a thorough discussion about the content of the resolution theorems in [37] and in [49], highlighting virtues and potential limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In Section 5, we consider the fundamental problem of existence of separatrices that, roughly speaking, concerns the existence of germs of analytic sets invariant by (germs of) singular foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The discussion is essentially conducted in (C3, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Expanding on the discussion of Section 3, we consider the existence of separatrices for codimension 1 foliation spanned by two commuting vector fields and state Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='3 affirmatively answering this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' We also detail the general strategy for proving this theorem which, in turn, emphasizes a few often overlooked points in resolution theorems for foliations as well as the major role played by the general question of “taming a core dynamics”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The second part of this section, we review some results on the existence of separatrices for foliations of dimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The nature of this second problem is far more topological/geometric and “core dynamics” plays a much smaller role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Yet, some of the results will find applications in the last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Finally, in Section 6 we discuss a particular class of singularities of foliations of dimension 1, namely those supporting semicomplete vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Albeit small in an appropriate sense, this class of singularities has rather distinguished properties and quite a few applications that make it worth studying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The section will precisely begin with proper definitions and a general discussion of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Once the basic setting is in place, we will go on to discuss two GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS 3 fundamental problems on the area: the first problem can vaguely be stated by asking how wild can the core dynamics be in this class of foliations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The main results here stem from the seminal paper of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Guillot about Halphen vector fields and their role in SL (2, C)-actions, see [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The second problem aims at quantifying how “degenerate” a singularity in this class can be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This second problem stems from a well known question raised long ago by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Ghys and the topic has applications in the study of automorphism groups of compact complex manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Acknowledgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' We are grateful to many of our colleagues for several discussion over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' A special thanks is due to D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Panazzolo who has helped us to understand many subtle points of his fundamental work on desingularization theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' We also thank J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Mattei for many discussions and explanations about the vast and fundamental work in singularity theory he has accomplished with his collaborators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Rebelo and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Reis were partially supported by CIMI through the project “Complex dynamics of group actions, Halphen and Painlev´e systems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Reis was also partially supported under the project “Means and Extremes in Dynamical Systems” with reference PTDC/MAT- PUR/4048/2021 and also by CMUP, member of LASI, which is financed by national funds through FCT Funda¸c˜ao para a Ciˆencia e Tecnologia, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', under the project UIDB/00144/2020 as also supported .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Basics in the local theory of foliations and the special case of dimension 2 It is convenient to begin by recalling the definition of 1-dimensional singular holomorphic foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' First, let X = f1∂/∂x1 + · · · + fn∂/∂xn be a non-trivial holomorphic vector field defined on an open set V of Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The singular set Sing (X) of X is then given by �n i=1{fi = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' It is a (proper) analytic subset of V and it is well known that Sing (X) has codimension 1 if and only if the coordinate functions fi admit a non-trivial common factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' We are then able to define singular holomorphic foliations as they will be considered through- out this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let M be a complex manifold and consider a covering {(Uk, ϕk)} of M by coordinate charts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' We denote by n the dimension of M so that ϕk(Uk) is an open set of Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let M and {(Uk, ϕk)} be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' A singular holomorphic foliation F of dimension 1 on M consists of a collection of holomorphic vector fields Yk satisfying the following conditions: For every k, Yk is a holomorphic vector field defined on ϕk(Uk) ⊂ Cn whose singular set has codimension at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Whenever Uk1∩Uk2 ̸= ∅, we have Yk1 = gk1k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (ϕk2◦ϕ−1 k1 )∗Yk2 for some nowhere vanishing holomorphic function gk1k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The singular set Sing (F) of a foliation F is then defined as the union over k of the sets ϕ−1 k (Sing (Yk)) ⊂ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Therefore the singular set of any holomorphic foliation has codimension at least two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, a foliation has no divisor of zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Conversely, we say that a holomorphic vector field Y is a local representative of the 1- dimensional foliation F if Y is tangent to F and the singular set of Y has codimension at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' It is clear that representative vector fields are locally unique up to multiplication by an invertible holomorphic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Analogously, we might also consider a differential 1-form ω on V ⊆ Cn, ω = g1dx1 + · · · + gndxn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Again the singular set Sing (ω) of ω is given by the intersection �n i=1{gi = 0} and it is an analytic set of V which has codimension 1 if and only if there is a non-trivial common factor for the functions g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' , gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Away from its singular points, the kernel of ω defines a distribution of complex hyperplanes on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' If in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='1 we replace “local vector fields” by “integrable local 1-forms”, we obtain the notion codimension 1 foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' More precisely, we have: 4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REBELO AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REIS Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let M be a complex manifold and {(Uk, ϕk)} a covering of M by coordinate charts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' A singular holomorphic foliation D of codimension 1 on M consists of a collection of differential 1-forms Ωk satisfying the following conditions: For every k, Ωk is a differential 1-form defined on ϕk(Uk) ⊂ Cn with singular set of codimension at least 2 and such that Ωk ∧ dΩk vanishes identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Whenever Uk1∩Uk2 ̸= ∅, we have Ωk1 = gk1k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (ϕk2◦ϕ−1 k1 )∗Ωk2 for some nowhere vanishing holomorphic function gk1k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular the singular set Sing (D) of D again has codimension at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The notion of representative 1-form for a codimension 1 foliation D is defined analogously to the notion of representative vector fields in the case of foliations with dimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Whereas our main focus here is on germs of foliations, or in slightly more concrete terms, on foliations defined on a neighborhood of the origin in Cn, the reader will notice that the global point of view considered in Definitions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='2 is really indispensable to investigate the local structure of the singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Indeed, globally defined foliations - and in particular the “global dynamical phenomenon” mentioned in the Introduction - will come to fore in the context of birational theory of foliations which is needed, for example, if one seeks to “resolve singular points”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' It is also convenient to complement the above definitions with a couple of comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Already on (Cn, 0), n ≥ 3, it is easy to see a first fundamental difference between foliations of dimension 1 and foliations of codimension 1 arising from the Frobenius condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' To formulate this, note that any choice of local holomorphic functions f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' , fn naturally gives rise to two (singular) distributions: one of lines and one of hyperplanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In fact, to the collection of functions f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' , fn, we may associate the vector field Y = f1∂/∂x1 + · · · + fn∂/∂xn or the 1-form Ω = f1dx1 + · · · + fndxn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Whereas the local integral curves of Y always yield a 1- dimensional foliation F, the Frobenius condition for Ω to yield a codimension 1 foliation is non-trivial and amounts to requiring the 3-form Ω ∧ dΩ to vanish identically which, in turn, leads to a highly non-trivial PDE system involving the functions f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' , fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In general, foliations of dimension 1 are very abundant, at least in algebraic manifolds, and they may have an extremely complicated dynamical behavior, more on this in Section 3, see also [24], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This contrasts with the case of codimension 1 foliations that are far more rigid and in several cases amenable to classification, at least at conjectural level, all codimension 1 foliations on, say, CPn should be transversely homogeneous or can be obtained as a suitable pull-back of a foliaion defined on a surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' For an interesting discussion of several of global aspects of codimension 1 foliations, we refer the reader to [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' A basic object in the local theory of foliations that has largely motivated its early development is the notion of separatrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Although the definition of separatrix depends on the dimension of the foliation, the cases of foliations of dimension 1 and of codimension 1 can naturally be formulated together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let F (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' D) be a foliation of dimension 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' codimension 1) on (Cn, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' A separatrix S for F (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' D) is the germ of an irreducible analytic set of dimension 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' codimension 1) containing 0 ∈ Cn and invariant by F (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Separatrices are objects of natural interest since they fit the framework of “invariant man- ifolds” in dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, their presence may enable one to reduce the di- mension of the phase space of the system in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Yet, in the local theory of foliations as discussed here, the notion of separatrix first appeared in the classical work of Briot and Bouquet [2] where it was claimed that every foliation on (C2, 0) admits at least one separatrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Much GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS 5 later, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Thom has sought to generalize the existence of separatrices for every codimension 1 foliation on (C3, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The first example of a codimension 1 foliation on (C3, 0) without separatrix was, however, found by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Jouanolou [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' As it will be seen in the next section, Jouanolou’s counterexample relies on the core dynamics of certain 1-dimensional foliations on (C3, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let us also point out that a gap in the arguments of Briot and Bouquet was later found and the existence of separatrices for foliations on (C2, 0) was firmly established by Camacho and Sad in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Another fundamental notion in the theory of singularities of foliations is the notion of eigen- values of a foliation at a singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' To abridge the discussion, for the time being let us restrict ourselves to the case of foliations of dimension 1 (see Section 4 for a more general discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Without loss of generality, it suffices to consider the case of foliations F of dimen- sion 1 defined on (Cn, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Given F as above, up to reducing the neighborhood of the origin, there is a holomorphic vector field Y whose zero-set has codimension at least 2 and such that F is nothing but the foliation induced by the local orbits of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' As mentioned, Y is said to be a representative of F and, while Y is not unique, two representative vector fields for the same foliation F differ by multiplication by an invertible holomorphic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let F be a 1-dimensional holomorphic foliation on (Cn, 0) and let Y denote a representative vector field for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Assume that F is singular at the origin, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', Y (0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Then the eigenvalues of F at 0 ∈ Cn are the eigenvalues of the Jacobian matrix D0Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Since Y is well defined up to multiplication by an invertible holomorphic function, there fol- lows that the eigenvalues of F at 0 ∈ Cn are well defined only up to simultaneous multiplication by a non-zero constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Blow-ups and dicritical singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Blow-ups are a standard tool to produce non- trivial birational maps and to understand the local structures of singular points, whether these are “singularities of the ambient space” or “singularities of a foliation on a smooth space”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The transform of a foliation under a blow-up map is called the blow-up of the foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The blown-up space, however, contains an exceptional divisor which may or may not be invariant by the transformed foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This issue gives rise to the notion of dicritical foliation at a singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let M be a complex manifold equipped with a holomorphic foliation H and consider a blow-up map π : � M → M centered at C ⊂ Sing(H), where Sing(H) stands for the singular set of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The foliation H is said to be dicritical with respect to π if its corresponding blow-up � H does not leave the exceptional divisor π−1(C) invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Whenever no misunderstanding is possible, we will simply say that a given foliation is, or is not, dicritical without specifically mentioning to the blow-up map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Also, for most of our discussion, it will suffice to consider blow-ups centered at single points (sometimes called one- point blow-ups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' For this type of blow-up, the characterization of 1-dimensional dicritical foliations is very simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' More precisely, let F be a 1-dimensional foliation on (Cn, 0) and fix a representative vector field Y of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Denote by Yk the non-zero homogeneous component of least degree in the Taylor series of Y based at 0 ∈ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Then, we have: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The foliation F is dicritical with respect to the blow-up centered at 0 ∈ Cn if and only if Yk is a multiple of the radial vector field R = x1∂/∂x1 + · · · + xn∂/∂xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' It suffices to compute the pull-back of Y in the coordinates (x1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' , un) for the blow- up of Cn where the blow-up map π is given by π(x1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' , un) = (x1, x1u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' , x1un), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' for example [23] or [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' □ 6 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REBELO AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REIS In more general terms, a foliation F is said to be dicritical at a center C if there exists a sequence of blow-ups beginning at C and leading to a foliation which does not leave all the irreducible components of the global exceptional divisor invariant (for details see Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Since a blow-up map is proper, and therefore so is a composition of blow-up maps, there follows from Remmert’s theorem that a leaf of the foliation �F transverse to (a component of) the exceptional divisor must project to a separatrix for the initial foliation F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Thus we obtain the following simple characteristic of 1-dimensional dicritical foliations: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' If the foliation F at the center C is dicritical, then the union of separatrices of F through points of C yields a set with non-empty interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' □ The converse to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='9 is known to hold for ambient spaces of dimension up to 3, and it is a simple consequence of “resolution theorems”, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Whereas the result is likely to hold in general, a proof of this statement dispensing with “resolution” results seems to be still lacking in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The above lemmas show that 1-dimensional dicritical foliations are, somehow, very special.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, a “generic foliation” is not dicritical at their singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Also, owing to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='9, for most of the problems discussed here involving 1-dimensional foliations, we can assume without loss of generality that the foliation in question is not dicritical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Examples of dicritical foliations are far more abundant when we consider codi- mension 1 foliation in ambient spaces of dimension 3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, it is unclear if there is any reasonable sense in claiming that a “generic foliation” is not dicritical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In fact, a good source of examples of dicritical foliations consists of exploiting the affine Lie algebra generated by a homogeneous polynomial vector field and by the radial vector field, see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Singularities of foliations on (C2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' As mentioned, singularities of foliations on (C2, 0) are the object of a highly developed theory, at least in the very general setting of non-dicritical foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In this paragraph, we shall collect some reasons that allowed so much progress in this topic and compare them with the situation of foliations on (C3, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (A) Seidenberg theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' It is commonly accepted that no general theorem in singularity theory can be proved without relying on a suitable “desingularization theorem”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In the theory of foliations, however, it is not possible in general to actually desingularize a foliation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', to obtain a non-singular model of the foliation up to birational transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In fact, whereas the phrase desingularization theorem is sometimes used as an abuse of language, a more accurate terminology would be reduction of singularities theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In other words, rather than looking for a non-singular foliation, we look for a foliation whose singular points are as “well behaved as possible”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Typically, we will look for a foliation all of whose singular points are elementary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', all of them have at least one eigenvalue different from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Seidenberg theorem [55] provides a suitable procedure to reduce the singularities of holo- morphic foliations on (C2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let F denote a singular holomorphic foliation defined on a neighborhood of (0, 0) ∈ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Seidenberg theorem asserts the existence of a finite sequence of blow-up maps, along with transformed foliations Fi (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' , n) F = F0 π1 ←− F1 π2 ←− · · · πn ←− Fn such that the following holds: Each blow-up map πi (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' , n) is centered at a singular point of Fi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' All singular points of Fn are elementary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', the foliation Fn possesses at least one eigenvalue different from zero at each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS 7 Denote by D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' , Dn the irreducible components of the total exceptional divisor associated with Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Each Di is therefore a rational curve with strictly negative self-intersection and the corresponding Dynkin diagram is a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (B) A global pseudogroup - Mattei-Moussu technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Assume next that the foliation F is not dicritical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Then, for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' , n, Di \\ Sing (Fn) is a regular leaf of Fn, where Sing (Fn) stands for the singular set of Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, all non-trivial dynamics associated with the foliation Fn is of transverse nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Moreover this transverse dynamics naturally arises from the holonomy representations of each of the leaves Di \\ Sing (Fn), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In turn, at least to a considerable extent, the dynamics of these representations can be merged together through the argument of “passage of corners” (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' “Dulac transform”), whenever Di ∩ Dj ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The preceding can be summarized by saying that all singular points of Fn are “dynamically connected” in the sense that their local dynamics blend together in a nice pseudogroup of maps of (C, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Furthermore, the dynamics of this pseudogroup encodes virtually all the information on the local structure of the initial foliation F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The method described above to investigate the singularities of foliations on (C2, 0) was very much set up in the seminal paper by Mattei and Moussu [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This technique has proven time and again to be extremely effective in a variety of situations in dimension 2, see [5] and [19] for two examples of problems whose solutions have involved this type of setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In the next section we will discuss how far this approach can be generalized to higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (C) Dynamics of pseudogroups acting on (C, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Although for many problems this issue plays a relatively minor role, let us still point out that the dynamics of pseudogroups acting on (C, 0) is itself a highly developed topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This type of dynamics was first investigate by Huddai- Verenov [22] and then by Il’yashenko in [24] where a “generic situation” of groups generated by hyperbolic diffeomorphisms was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In contrast, in [35], the authors have dealt with subgroups all of whose orbits are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' An absolute breakthrough then came with the works of Shcherbakov and of Nakai about general non-solvable subgroups, see [53], [54], [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The reader may consult [42] and references therein for a more complete account of these dynamics in the non-solvable case whereas solvable pseudogroups are discussed in detail in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' It should be pointed out that much progress in terms of construction of moduli spaces for foliations on (C2, 0) and in describing the topology of leaves has been made in recent years, chiefly by Mar´ın, Mattei, and Salem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' While these aspects will not be discussed in this survey which is mostly devoted to higher dimensional situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Yet, the reader interested in the topology of leaves will find more up-to-date information in [30], [31], and [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' As to the construction of moduli spaces, we refer to [32] and to the preprints [33] and [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Splitting the problem: core dynamics and resolution The main object of this section are 1-dimensional foliations defined around the origin of Cn, for n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Yet, most of the discussion can be conducted without loss of generality in the case n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' It is useful to begin by recalling some well known facts about foliations on complex projective spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let CPn be viewed as the space of radial lines through the origin of Cn+1 and denote by Π : Cn+1 \\ {0} → CPn the canonical projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Also, for λ ∈ C∗, denote by hλ : Cn+1 → Cn+1 the homothety defined by hλ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' , xn+1) = (λx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' , λxn+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Finally let R denote the radial vector field R = x1∂/∂x1 + · · · + xn+1∂/∂xn+1 and consider a homogeneous polynomial vector 8 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REBELO AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REIS field X = P1 ∂ ∂x1 + · · · + Pn+1 ∂ ∂xn+1 of degree d on Cn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In other words, each Pi is a degree d homogeneous polynomial, for every i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' , n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In what follows X is always assumed to satisfy the following conditions: (1) The singular set of X on Cn+1 has codimension at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (2) The vector fields X and R are linearly independent at generic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Next note that we have h∗ λX = λd−1X so that the vector fields h∗ λX and X are everywhere parallel for any fixed value of λ ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, if p ∈ Cn+1 is a point at which X(p) and R(p) are linearly independent, then X(p) induces a direction in Tq=Π(p)CPn which is well defined in the sense that it does not depend on p ∈ Π−1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' From this, it easily follows that X induces a singular holomorphic foliation F on CPn in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' A standard application of Serre’s GAGA principle yields a type of converse for the above construction, namely the following proposition holds, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' for example [23], [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let F denote a singular holomorphic foliation on CPn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Then, there exists a homogeneous polynomial vector field X on Cn+1 having singular set of codimension at least 2 and inducing the foliation F on CPn by means of the above described construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' □ Whereas, given F, the mentioned homogeneous vector field X of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='1 is not uniquely defined, two homogeneous polynomial vector fields having singular set of codimension at least 2 and inducing the same foliation on CPn must have the same degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Thus we can talk about the degree of a foliation on CPn as follows: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The degree of a foliation F on CPn is the degree of a homogeneous polynomial vector field on Cn+1 having singular set of codimension at least 2 and inducing F in CPn viewed as the space of radial lines of Cn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Naturally blow-ups provide an alternative way to realize the foliation induced on CPn by a homogeneous polynomial vector field X on Cn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let �Cn+1 stand for the blow-up of Cn+1 at the origin and consider a homogeneous vector field X as above on Cn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The blow up � X of X induces on �Cn+1 the blow up �F of the the foliation F induced by X on Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Since, by assumption, X is not everywhere parallel to the radial vector field R, there follows that the foliation �F leaves invariant the exceptional divisor π−1(0) ≃ CPn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The restriction of �F to the exceptional divisor π−1(0) ≃ CPn can then naturally be identified with the foliation induced by X on CPn - viewed as the space of radial lines of Cn+1 - by means of the preceding construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Note that the blow up construction does not really requires the vector field to be homoge- neous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In fact, as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='8, consider a holomorphic vector field Y defined around the origin of Cn+1 whose Taylor series takes on the form Y = �∞ i=k Yi, where Yi stands for the homogeneous component of degree i of this Taylor series and Yk is not identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' As in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='8, we assume that Yk is not everywhere parallel to the radial vector field R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The blow up of Y induces a holomorphic foliation �F on a neighborhood of the exceptional divisor π−1(0) ⊂ �Cn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Moreover, since Yk is not a multiple of R, this foliation leaves π−1(0) ≃ CPn invariant and, in addition, it is immediate to check that the restriction of �F to π−1(0) coincides with the restriction to π−1(0) of the foliation induced on �Cn+1 by the blow up of Yk (alone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, the restriction of �F to π−1(0) is identified with the foliation induced by the homogeneous vector field Yk on CPn viewed as the space of lines of Cn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The preceding motivates the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS 9 Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let F be a 1-dimensional holomorphic foliation defined around the origin of Cn and assume that the blow up �F of F at the origin leaves the exceptional divisor π−1(0) invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Then the foliation induced on CPn−1 ≃ π−1(0) by the restriction of �F is called the core foliation of F and its global dynamics is referred to as the core dynamics of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Again, if Y is a representative of F and Yk is as above (Y = �∞ i=k Yi), the preceding then shows that the core foliation of F is nothing but the foliation induced on CPn−1 by the homogeneous vector field Yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 1-dimensional foliations and dicritical codimension 1 foliations on C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The pre- ceding discussion about foliations on projective spaces also applies to the case of codimension 1 dicritical foliations on (Cn, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' To be more precise, codimension 1 dicritical foliation D on (Cn, 0) also induces through the one-point blow-up centered at the origin a foliation on CPn−1, that will also be called the core foliation of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' It is fair to say that this phenomenon and the corresponding dynamics were first exploited by Jouanolou [26] in his famous counterexample to a question posed by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Thom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' We shall review this issue below and go somewhat further by exploiting the results in [28] to see how difficult the situation may become.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In the sequel, we set n = 3 to abridge notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' First, let us characterize codimension 1 foliations that are dicritical for the blow-up of C3 centered at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Since the lemma below does not seem to be accurately stated in the literature, a detailed - albeit straightforward proof is included below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Assume that D is a singular codimension 1 foliation defined on (C3, 0) and denote by �D its blow-up centered at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Then the exceptional divisor π−1(0) ≃ CP2 is invariant under �D if and only if no holomorphic vector field Z tangent to D admits a first non-zero homogeneous component (at the origin) that is a multiple of the radial vector field R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let D be given by a holomorphic 1-form Ω = F dx + G dy + H dz whose singular set has codimension at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Denote by Ωk the first non-zero homogeneous component of Ω at the origin, where k stands for the degree of Ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Next, let Ωk = F kdx + Gkdy + Hkdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' A direct inspection shows that π−1(0) is not invariant by �D if and only if (1) xF k + yGk + zHk = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Now, note that if Z is any vector field tangent to D, and whose first non-zero homogeneous component is denoted by Zl, then Zl naturally provides a solution for {Ωk = 0}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', we have Ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='Zl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' However, if Zl happens to be a multiple of the radial vector field, then Ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='Zl = 0 is tantamount to Equation (1) which is thus satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Hence, the exceptional divisor is not invariant by �D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' To show that the existence of a vector field Z satisfying the above mentioned conditions is also necessary, we proceed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Assume that D is dicritical, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', that Equation (1) holds and denote by ∧ the standard exterior power of two vectors on C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Next define a vector field v by letting v(p) = R(p) ∧ (F(p), G(p), H(p)) and then set Z(p) = v(p) ∧ (F(p), G(p), H(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Clearly Z is tangent to the foliation D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' To complete the proof of the lemma, it suffices to check that the first non-zero homogeneous component of Z at the origin is a multiple of the radial vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' For this, note that we have (2) Z = (zFH + yFG − x(H2 + G2) , xFG + zHG − y(F 2 + H2) , yHG + xFH − z(G2 + F 2)) In particular the order of Z at the origin is at least 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The homogeneous component of degree 2k + 1 is, in turn, given in vector notation by (xF k + yGk + zHk)(F k, Gk, Hk) − ((F k)2 + (Gk)2 + (Hk)2)(x, y, z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 10 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REBELO AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REIS In view of Equation (1), we conclude that the component of degree 2d + 1 of Z at the origin is given by −((F k)2 + (Gk)2 + (Hk)2)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' To finish the proof of the lemma it suffices to show that the polynomial (F k)2+(Gk)2+(Hk)2 cannot vanish identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This, however, can easily be done by using the variables (x, t, u) where the blow-up map becomes Π(x, t, u) = (x, xt, xu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In these variables, the dicritical condition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Equation (1)) means that F k(1, t, u) + tGk(1, t, u) + uHk(1, t, u) must vanish identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Now, suppose for a contradiction that (F k)2 + (Gk)2 + (Hk)2 is also identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Then the two equations taken together imply that (t2 + 1)(Gk)2 + 2tu(Gk)(Hk) + (u2 + 1)(Hk)2 vanishes identically as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' By solving the corresponding last equation for Gk, we derive a contradiction with the fact that Gk is itself a polynomial in the variables t, u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The lemma is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' □ Next, let us consider again a homogeneous polynomial vector field X on C3 satisfying condi- tions (1) and (2) in the previous subsection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' the singular set of X has codimension at least 2 and the vector fields X and R are linearly independent at generic points (note that in the case of homogeneous vector fields of degree at least 2, conditions (1) and (2) are equivalent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Since X is homogeneous, we have [R, X] = (d − 1)X where d stands for the degree of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Thus the pair X and R generates the Lie algebra of the affine group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular the distribution of planes (of dimension 2) spanned by X and R is involutive and hence integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let us then denote by D the codimension 1 foliation spanned by X and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let �D stands for the blow-up of D centered at the origin so that �D is defined on �C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Owing to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='4, the exceptional divisor π−1(0) ≃ CP2 is not invariant under �D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Furthermore, the structure of the foliation �D (and hence that of D) is essentially as complicated as the structure of the blow-up �F of F, where F denotes the foliation induced by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This observation deserves further comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' To begin with, recall that �C3 can also be seen as the tautological line bundle over π−1(0) ≃ CP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The bundle projection will be denoted by �Π : �C3 → π−1(0) since it can naturally be identified with the canonical projection Π : C3 \\ {(0, 0, 0)} → CP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Next, recall that, unlike �D, the foliation �F leaves the exceptional divisor invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, at a point p of π−1(0) ≃ CP2 that is regular for �F, this foliation defines a direction up ∈ Tpπ−1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Next, for p “sufficiently generic”, the leaf of �D intersects transversely π−1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This transverse intersection naturally defines a direction vp ∈ Tpπ−1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' It is immediate to check that the directions of vp coincides with the one defined by �F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Denoting by �F|π−1(0) the foliation on π−1(0) obtained by restriction of �F, we have the following: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The leaves of the foliation �D are of the form �Π−1(L) where L is a leaf of �F|Π−1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Similarly, every leaf of �D is invariant by the foliation �F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' □ Recalling that every foliation on a projective space is induced by a homogeneous polynomial vector field, the interest of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='5 is actually captured by the following slightly loose state- ment: every foliation on CP2 is naturally the core foliation for singularities of both dimension 1 and codimension 1 foliations on (C3, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Before considering some concrete applications of the previous remark, let us close this sec- tion by pointing out that the above construction allows us to define the core of a dicritical codimension 1 foliation on (C3, 0) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let D be a codimension 1 holomorphic foliation defined around the origin of C3 and assume that the blow up �D of D at the origin does not leave the exceptional divisor GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS 11 π−1(0) invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Then the foliation induced on CPn−1 ≃ π−1(0) by the restriction of �F is called the core foliation of F and its global dynamics is referred to as the core dynamics of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Jouanolou’s example, chaotic dynamics, and their meaning for singularity the- ory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let us go back to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Thom’s question on the existence of separatrices for codimension 1 foliations on (C3, 0), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' As pointed out in Section 2, it is not always easy to construct codimension 1 foliations due to Frobenius integrability condition that has to be satis- fied by the distribution of planes in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Yet, the discussion revolving around Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='5 also indicates a simple way to construct lots of dicritical codimension 1 foliations on C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' More precisely, every foliation on CP2 yields one such dicritical codimension 1 foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let then D be a dicritical codimension 1 foliation as above and assume that D admits separa- trices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let then S denote a germ of an irreducible separatrix for D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Since S has codimension 1, there follows the existence of a germ of an irreducible holomorphic function f : (C3, 0) → (C, 0) such that S coincides with the set {f = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In terms of Taylor series, we set f = �∞ i≥l fi where l is the degree of the first non-zero homogeneous component of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let C ⊂ CP2 be the curve defined on the projective plane by the homogeneous equation {fl = 0} (the tangent cone to S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' If we denote by F the core of D (recall than that F is a 1-dimensional foliation on CP2), then the following can be said: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' With the preceding notation, the curve C ⊂ CP2 is invariant by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The foliation D is defined by an integrable 1-form Ω whose Taylor series takes on the form Ω = �∞ i=k Ωi where k stands again for the first non-zero homogeneous component of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' A simple argument based on degrees shows that the 1-form Ωk is integrable as well, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', it satisfies Frobenius equation Ωk ∧ dΩk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Similarly, one checks that the (homogeneous) surface defined by {fl = 0} yields a separatrix for the codimension-1 foliation Dk induced by Ωk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Set Ωk = F kdx + Gkdy + Hkdz and, as usual, let R denote the radial vector field on C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Next recall that a homogeneous vector field of C3 representing F is well defined only up to a multiplicative constant and addition of a multiple of the radial vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Now, since Ωk is homogeneous, the vector R(p) is contained in the plane defined by the kernel of Ωk(p) at the point p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Hence, up to eliminating multiplicative factors, a representative vector field X for F can be obtained by letting X(p) = R(p) ∧ (F k(p), Gk(p), Hk(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular X(p) lies in the kernel of Ωk(p), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', X is tangent to the foliation Dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Finally, since at regular points p ∈ {fl = 0} the tangent space at {fl = 0} and the kernel of Ωk(p) coincide, we conclude that X is tangent to the surface {fl = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The lemma then follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' □ In view of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='7, the basic remark of Jouanolou concerning Thom’s conjecture was the following one: if we can find a foliation F on CP2 leaving invariant no algebraic curve, then the (dicritical) codimension 1 foliation D arising from combining the radial vector field of C3 and a representative of homogeneous vector field for F will admit no separatrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Jouanolou’s remark is possibly the first instance where the existence of the core dynamics actually impacts the study of singularities of foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' From this point of view, the main result of Jouanolou in [27] can be stated as follows: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [27] For every d ≥ 2, the foliation induced on CP2 by the vector field Xd = yd∂/∂x + zd∂/∂y + xd∂/∂z leaves no algebraic curve invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Jouanoulou theorem implies, in particular, that for every fixed d ≥ 2, there exist foliations of degree d that are not tangent to any algebraic curve of CP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 12 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REBELO AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REIS Armed with the above theorem, there follows from what precedes that the codimension 1 Jouanolou foliation Jd, d ≥ 2, of C3 which is defined as the singular foliation spanned by Xd and the radial vector field is a counterexample to Thom’s question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The well-known explicit 1-form Ω, Ω = (yxd − zd+1) dx + (zyd − xd+1) dy + (xzd − yd+1) dz , defining the foliation Jd can promptly be obtained by taking the vector product of Xd and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The next question is to wonder how far the core dynamics can influence the study of singu- larities of foliations, say of dimension 1 on Cn, n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In other words, owing to the discussion in this section, the detailed understanding of the local structure of one such foliation arguably passes through the global description of its core foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This understanding would require, in particular, a (global) control of the dynamics of the core foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' At this point, we might wonder whether it is possible to obtain such an accurate local description of all 1-dimensional foliations on, say, (C3, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' From the standpoint emphasized above, an easier question would be to provide a reasonable global description of all or nearly all foliations on CP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Unfortunately, the latter question does not seem to admit an affirmative answer as it follows from Loray-Rebelo theorem [28] as stated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Fix positive integers n and d, with min{n, d} ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' A straightforward counting of parameters shows that the space Fol(d)(n) CP of degree d foliations on CPn can be identified with a Zariski-open set of the complex projective space of dimension (d + n + 1)(d + n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This space of foliation can then be furthered moduled out by the action of the automorphism group PSL (n + 1, C) of CPn but this will not be needed in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The main upshot here is that Fol(d)(n) CP can be parameterized by a finite dimensional complex manifold and, in particular, inherits of a natural topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' With this notation, the main result of [28] reads as follows: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [28] Fixed n, d ≥ 2, there exists a non-empty open subset U ⊂ Fol(d)(n) CP such that every foliation F lying in U satisfies all the conditions below: (1) All singular points of F are hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, they form a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (2) Minimality: Every leaf of F is dense in CPn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (3) Ergodicity: Every measurable set of leaves has either zero or total Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (4) Rigidity: If F′ ∈ Fol(d)(n) CP is conjugate to F by a homeomorphism h : CPn → CPn that is close to the identity, then F and F′ are also conjugate by an element of PSL (n + 1, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The level of dynamical complication exhibited by the foliations indicated above puts any ac- curate description of them basically out of reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Moreover, even up to topological conjugation, it is not possible to achieve a reasonable list of “models” or “normal forms” owing to the above indicated rigidity phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' It is convenient to expound a bit on the consequences of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='9 from the point of view of singularity theory for 1-dimensional foliations on dimensions 3 and greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Consider then a foliation lying in the set U ⊂ Fol(d)(n) CP provided by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' As a foliation defined on CPn, it can be represented by some homogeneous polynomial vector field X on Cn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In other words, if F is the foliation on Cn+1 induced by the local orbits of X then �F, its (one-point) blow-up at the origin, leaves the exceptional divisor π−1(0) ≃ CPn invariant and is such that the restriction of �F to π−1(0) ≃ CPn is naturally identified with the initial foliation in U ⊂ Fol(d)(n) CP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Now recall that the vector field X is not uniquely defined: most notably, we can add to X any multiple of the radial vector field by a homogeneous polynomial of degree d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Since, in GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS 13 addition, the singularities of the initial foliation in U are all hyperbolic, it is easy to conclude that the vector field X can be chosen so as to fulfill the following conditions: (1) The foliation F has an isolated singularity at the origin of Cn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (2) The foliation �F, viewed as foliation on a manifold of dimension n + 1, still have only hyperbolic singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Furthermore, a generic choice of the initial foliation in U and of the vector field X allows us to rule out the existence of resonances at the singular points of �F as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Thus, all the singularities of �F are, in fact, linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Also, all the above mentioned characteristic are stable under higher order perturbations of a representative vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The situation can then be summarized as a statement in itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (Corollary of [28]) For every degree d ≥ 2, there exists a non-empty open set V of homogeneous vector fields of degree d in X(Cn+1, 0) such that every germ of foliation F represented by a holomorphic vector field X having the form X = Xd +h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', with Xd ∈ V and where h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' stands for higher order terms, satisfy all of the following conditions: (1) The one-point blow up �F of F at the origin leaves the exceptional divisor π−1(0) ≃ CPn invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (2) All singular points of �F are hyperbolic and linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, �F has exactly dn+1 − 1 d − 1 singular points and all of them lie in π−1(0) ≃ CPn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (3) The restriction of �F to π−1(0) ≃ CPn defines a degree d foliation of CPn lying in the open set U ⊂ Fol(d)(n) CP given by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let us point out that the formula in item (2) for the number of singular points of �F, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', for a degree d foliation on CPn all of whose singular points are hyperbolic is well known and can be proved in a variety of ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' For example, by choosing affine coordinates yielding a “hyperplane at infinity” on which the foliation has no singular point and then applying B´ezout theorem to the corresponding polynomial vector field representing the foliation in the above indicated affine coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Naturally the content of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='10 can be adapted to germs of codimension 1 dicritical foliations on (C3, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' To close this section, it is convenient to make a parallel with the discussion in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='2 for singularities of foliations on (C2, 0) so as to better appreciate the difficulties arising from the existence of wild core dynamics as stated in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (A’) Generalizations of Seidenberg theorem to (Cn, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The problem is wide open for n ≥ 4 though sharp desingularization theorems are now established for n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The topic is of undisputed interest since virtually all general statements about singularities rely, directly or indirectly, on a suitable “resolution theorem”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Yet, for n ≥ 3, the ability to obtain a model of the foliation where all singular points are “simple enough” might still be a long way off of providing an accurate description of the singularity in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' To substantiate the above claim, it suffices to consider the local foliations F on (Cn, 0) provided by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The blow-up �F of F at the origin provides a birational model for F possessing only “simple singular points”: in fact, all singularities of �F are hyperbolic and linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In other words, the local behavior of �F around each of its singular points is essentially trivial and promptly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The very complicated dynamical behavior of F 14 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REBELO AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REIS around the origin is, however, encoded in its core dynamics (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='5) but the global nature of the core dynamics prevents resolution theorems to yield any insight into this dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (B’) Taming the core dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' If one is to fully understand the structure of a foliation around a singular point, then an accurate description of its core dynamics needs to be envisaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' If Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='10 tells us this is a kind of unrealistic goal, it also raises the question of “selecting” those classes of singular points allowing a more detailed description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This is a very interesting point as it hints at considering the connections between singularity theory and the remainder of Mathematics or, even, Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Singularities playing a special role in problems from Geometry, Complex Analysis and/or Integrable Systems are likely to be amenable to a more complete analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Examples of these situations will be discussed in the forthcoming sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In terms of “taming core dynamics”, of course the ideal situation would be to have a core foliation defining an “integrable system” in some suitable sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Alternatively, for a number of problems, it might be enough to ensure the existence of (“sufficiently many”) algebraic invariant curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' An important issue involving invariant curves is that more often than not the dynamics of the foliation in question can be investigated in more details on a neighborhood of them, especially when their fundamental group contains more than a single generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This study, whereas of more global nature, is somehow akin to “Mattei-Moussu pseudogroup technique” mentioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Interesting examples where this point of view have successfully been employed - even outside the scope of singularity theory - include [24], [28], and [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (C’) Dynamics of pseudogroups acting on (Cn, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The perspective of focusing in the local dynamics arising from the holonomy of an invariant algebraic curve in higher dimensions naturally leads us towards considering the dynamics of subgroups of Diff (Cn, 0), n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' As was to be expected, many new dynamical phenomena arise for n ≥ 2 compared to the situation n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' As pointed out in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='2 much is known about the dynamics of subgroups of Diff (C, 0) whereas for subgroups of Diff (Cn, 0), the theory is still in its early stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Nonetheless, we mention that generalizations to higher dimensions of Mattei-Moussu’s theo- rem on groups with finite orbits is by now well understood, see [46], [48], [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' These results are likely to have impact in problems about existence of first integrals but they might also provide insight in higher dimensional versions of the so-called “analytic limit set”, see [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Finally a major issue in the theory is to find sharp conditions to extend to higher dimensions the Shcherbakov-Nakai theory of local vector fields in the “closure of the group” [53], [54], [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Very little is known about this question aside from some results in [28] which rely on the existence of a hyperbolic contraction for the group in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This assumption looking rather far from sharp, the topic appears to be ripe for significant progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Resolution theorems in dimension 3 In the remainder of this survey we will discuss relatively recent progress in some of the several aspects of singularity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This section is devoted to “resolution theorems” while the next two sections will basically review the general problem of invariant varieties and the study of a particular and important class of singular points, namely the semicomplete ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In the course of these discussions theorems providing - at various degrees - some control on the core dynamics in question will play a prominent role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' As previously indicated, theorems on reductions of singular points are always of paramount importance in the theory whether or not there are major difficulties lying out of their reach (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' complicated core dynamics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' For foliations defined on complex 2-dimensional manifolds GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS 15 (or varieties), Seidenberg’s theorem provides a sharp reduction of singularities theorem (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' “resolution theorem”) that is particularly easy to manipulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Beyond dimension 2, decisive results exist only in dimension 3, where it is already necessary to distinguish between foliations of dimension 1 and foliations of codimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This section is devoted to reviewing and explaining the main “resolution theorems” for 1-dimensional foliations in dimension 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Owing to the classical Hironaka resolution theorem, we can assume that our singular foliations are always defined on manifolds (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' smooth complex spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Furthermore, since the problems are local, we may assume them to be defined on a neighborhood of the origin of Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The case n = 2 being settled by the above mentioned theorem of Seidenberg, we assume from now on that n = 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', our foliations are defined on a neighborhood of the origin of C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Working on (C3, 0), we need to distinguish between foliations of dimension 1 and foliations of codimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The case of codimension 1 foliations was settled earlier in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' However, the story involving foliations of dimension 1 - the main object of this survey - is longer and more elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Resolution results for foliations of dimension 1 on (C3, 0) have first appeared in [6], where the author proves the existence of a formal local uniformization theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In this work, the author also hints at the existence of a new phenomenon involving singularities possessing a certain formal separatrix (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' a formal curve invariant by the foliation) which posed serious difficulties to resolve the singularity by means of standard blow ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The issue was made clear by Sancho and Sanz who provided explicit examples of foliations in (C3, 0) that cannot be reduced by sequences of standard blow-ups centered at sets contained in the singular loci of the initial foliations and its transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' After the examples found by Sancho and Sanz, the next truly major result in the area is due to D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Panazzolo [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In [39], Panazzolo considers singularities of real foliations in (real) dimension 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' He works in the real setting, rather than in the complex one, mostly due to the fact that his original motivation lied in Hilbert’s problem about the number of limit cycles of a polynomial vector field on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In his work, Panazzolo shows that the corresponding germs of foliations can always be turned into a foliation all of whose singular points are elementary by means of a finite sequence of weighted blow ups centered at singular sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The proof is constructive and actually provides an algorithm to obtain the desired reduction of singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Later, relying on Panazzolo’s algorithm introduced in [39], McQuillan and Panazzolo were able to provide a very satisfactory answer to the generalization of Seidenberg’s theorem for foliations on (C3, 0) in [36], [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The preprint [36] was made available in 2007 and a few years later, Cano, Roche, and Spivakovsky revisited the topic from the point of view of valuation theory, see [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Their strategy is in line with Zariski’s general approach to desingularization problems and, hence, is essentially divided in two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' First, for a given foliation, we seek to “simplify” only the singularities lying in the center of a given valuation (identified with its transforms, or extensions, through blow-ups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Resolution results for singularities lying in the center of a valuation are often referred to as local uniformization theorems and the first part of Zariski approach aims at obtaining this type of statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Once a convenient local uniformization result is obtained, the second part of Zariski approach deals with its globalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' More precisely, once it is proved that for every valuation ν, the singularities lying in the center of ν can be simplified (in some appropriate sense), we try to conclude that, in fact, all singularities of the foliation can simultaneously be simplified in the same sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' When it comes to applying this point of view to singularities of foliations most of the difficulties related to the globalization procedure are handled pretty well a very general (axiomatic) gluing theorem due to O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Piltant [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Owing to Piltant’s theorem, 16 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REBELO AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REIS it is fair to say that the fundamental difficulty of resolution problems for foliations, in arbitrary dimensions, revolves around obtaining suitable local uniformization theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In view of what precedes, the content of [9] can roughly be summarized by claiming the exis- tence of a birational model for the initial foliation where the singular points are log-elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The reader is referred to [9] for the definition of log-elementary singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' For our pur- poses, it suffices to know that such singularities are, at worst, quadratic in the sense that they are locally given by a representative vector fields with non-zero second-jet at the singular point in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' One of the goals of [49] was to complete the work of Cano-Roche-Spivakovsky by deriving “final models” similar to those of [37], in order to obtain a global resolution theorem comparable to [36], [37] through Zariski classical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' We will compare the resolution theorems for foliations obtained by McQuillan-Panazzolo in [37] and by ourselves in [49], they correspond to Theorem 2 and Theorem A of the respective papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, it will be seen that in the context of foliations the two results are pretty much equivalent and can be summarized by the following assertion: given a singular holomorphic 1-dimensional foliation F on (C3, 0), there exists a birational model of F where all singular points are elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In this sense, the only difference between the theorems in question will be down to the way in which the desired birational model is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Persistent nilpotent singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' As already mentioned, Sancho and Sanz have showed the existence of foliations in (C3, 0) that cannot be reduced by sequences of standard blow-ups with centers contained in the singular set of the initial foliation and its transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In fact, their result is slightly more general in the sense that we may allow for blow-ups of invariant centers not necessarily contained in the singular locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' As a matter of fact, they have pro- vided a 3-parameter family of foliations whose elements cannot be turned into a foliation all of whose singularities are elementary by means of blow ups centered in the singular loci and whose generic element cannot be turned into a foliation with elementary singular points even if invariant centers are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This family of foliations is represented by the family of vector fields Xα,β,λ taking on the form (3) Xα,β,λ = x � x ∂ ∂x − αy ∂ ∂y − βz ∂ ∂z � + xz ∂ ∂y + (y − λx) ∂ ∂z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Accordingly, foliations in this family will be denoted by Fα,β,λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The foliations Fα,β,λ are nilpo- tent at the origin in the sense that so are the vector fields Xα,β,λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' For reference, it is convenient to make accurate the notion of nilpotent foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' A (1-dimensional) holomorphic foliation is said to have a nilpotent singularity at a singular point p if its representative vector field around p has non-zero nilpotent linear part at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The above notion of nilpotent singularity is well defined since it does not depend on the choice of the representative vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Also, whenever no misunderstanding about the singular point in question is possible, we will abridge notation by simply saying that F is a nilpotent foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Going back to the nilpotent foliations Fα,β,λ, we note that the plane {x = 0} is invariant by them and that it contains the singular set of Fα,β,λ which coincides with the axis {x = y = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Recalling that a singular point is said to be elementary if the representative vector field possesses at least one eigenvalue different from zero, we now have the following: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The foliations in the family Fα,β,λ cannot be turned into a foliation all of whose singular points are elementary by means of a sequence of standard blow-ups with centers contained in singular sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS 17 Sketch of Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Consider the one-point blow-up centered at the origin of (C3, 0) and let π stands for the blow-up map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let then (x, u, v) be the affine coordinates for the blown-up space where y = ux and z = vx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The pull-back π∗Xα,β,λ of the vector field Xα,β,λ is given by π∗Xα,β,λ = x � x ∂ ∂x − (α + 1)u ∂ ∂u − (β + 1)v ∂ ∂v � + xv ∂ ∂u + (u − λ) ∂ ∂v , whose expression is similar to the expression of Xα,β,λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In fact, the main difference between the two expressions concerns the last term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Note, however, that the origin of the present coordinates is not contained in the singular set of the induced foliation, which is given by {x = 0, u = λ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Thus, if we consider the translation T(x, u, v) = (x, u + λ, v + µ), the pull-back of π∗Xα,β,λ through T is given by x � x ∂ ∂x − (α + 1)u ∂ ∂u − (β + 1)v ∂ ∂v � + x(v + µ − λ(α + 1)) ∂ ∂u + (u − µ(β + 1)x) ∂ ∂v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In the particular, if we choose µ = λ(α + 1), the vector field in question coincides with the vector field Xα+1,β+1,λ(α+1)(β+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In other words, the transformed foliation of Fα,β,λ contains a nilpotent singular point belonging to the (initial) Sancho-Sanz family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' It can be checked that the same issue occurs if the blow-up centered at the curve of singular points of Fα,β,λ is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' □ Summarizing what precedes, every sequence of blow ups as above applied to a foliation in Sancho-Sanz family lead to a foliation having a singular point where the foliation is locally conjugate to another foliation in the initial family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, all transformed foliations will exhibit a nilpotent singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This nilpotent singular point has a geometric interpretation naturally related to the issues raised by Cano in [6] for a resolution by standard blow-ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In fact, by elaborating in the above indicated argument, Sancho and Sanz have shown that the parameters α, β, λ can be chosen so that the foliation associated with the vector field Xα,β,λ possesses a strictly formal separatrix S = S0 through the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Moreover, given a sequence of blow ups as before, the sequence of points {pn} in the exceptional divisors corresponding to the position of the (persistent) nilpotent singularity is determined by the sequence of transforms {Sn} of the formal separatrix S = Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' We should still note that, the fact that every separatrix Sn is stricty formal says that even in the case we allow blow-ups to be centered at analytic invariant curves that are not necessarily contained in singular set of the foliation, a resolution procedure still does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In terms of the relation between foliations and - possibly formal - separatrices, a natural object that plays an important role is the notion of multiplicity of the foliation along the separatrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let X be a representative vector field of F and ϕ the Puiseux parametrization of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let ϕ∗X|S stands for the pull-back of the restriction of X to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' If ϕ∗X|S = g(t)∂/∂t, then the multiplicity of F along S is defined as the order of g at 0 ∈ C (assuming that ϕ(0) coincides with the singular point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In [49], we introduced the notion of persistent nilpotent singular point which is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' A nilpotent singular point p0 of a foliation F0 is said to be persistent if there exists a formal separatrix S0 for F0 through p0 such that for every sequence of blowing-ups F0 π1 ←− F1 π2 ←− · · · πl ←− Fn where Fi stands for the transformed of Fi−1 through the (standard) blow-up centered at some Ci−i ⊆ Sing (Fi−1) containing the point pi−1 (selected by the transformed separatrix Si−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' in the sense that it corresponds to the intersection of Si−1 with the excetional divisor),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' the following conditions are satisfied 18 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REBELO AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REIS (a) the singular points pi are all nilpotent singular points for the corresponding foliations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (b) the multiplicity of Fi along Si does not depend on i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The multiplicity of a foliation F along a separatrix (possibly a formal one) S is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let X be a representative vector field of F and ϕ the Puiseux parametrization of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let ϕ∗X|S stands for the pull-back of the restriction of X to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' If ϕ∗X|S = g(t)∂/∂t, then the multiplicity of F along S, mult(F, S), is defined as the order of g at 0 ∈ C (assuming that ϕ(0) coincides with the singular point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The role played by the notion of multiplicity of a foliation along a separatrix is closely related to its natural behavior under blow ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Recall that the order of a foliation at a singular point is nothing but the order of a representative vector field X , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' the degree of the first non-zero jet of X, at the singular point in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' With this notation, assume that �F is obtained by blowing up F at a singular point p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Assume also that S is a (formal) separatrix of F at p and denoted by �S the transform of S which yields a (formal) separatrix for �F at the point �p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Then we have: (4) mult( �F, �S) ≤ mult(F, S) with equality holding if and only if the order of F at p equals 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The same formula holds for blowing ups centered at a curve contained in the singular set of F, up to considering a variant of the notion of “order of the foliation” that is adapted to the center of the blow up, for details see [49] or the discussion at the end of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' It is easy to understand the interest of the multiplicity of a foliation F along a separatrix from the above perspective: if the existence of a (formal) separatrix S is ensured, then its multiplicity will drop strictly providing that the order of F at the singular point in question is greater than or equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Moreover, once this decreasing sequence stabilizes, then the corresponding singular point is either elementary or nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' From this it also follows that it is useful to understand persistent nilpotent singular points in order to establish resolution theorems for foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Clearly, in dimension 2, persistent nilpotent singular points do not exist as follows from Seidenberg theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In dimension 3, their existence is established by the above discussed examples due to Sancho and Sanz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' A characterization of these points in dimension 3 in terms of normal forms can be formulated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [49] Assume that F cannot be resolved by a finite sequence of standard blow- ups centered at singular sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Then there exists a sequence of one-point blow ups (centered at singular points) leading to a foliation F′ with a singular point p around which F is given by a vector field of the form (y + zf(x, y, z)) ∂ ∂x + zg(x, y, z) ∂ ∂y + zn ∂ ∂z for some n ≥ 2 and holomorphic functions f and g of order at least 1 with ∂g/∂x(0, 0, 0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Furthermore we have: (1) The resulting foliation F′ admits a formal separatrix at p which is tangent to the z-axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (2) The exceptional divisor is locally contained in the plane {z = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='4 deserves a couple of comments as it has a natural analogue in [37], namely: In [37], the authors obtain an alternative characterization of persistent nilpotent singu- larities which are presented as singular points arising from elementary ones by means of a Z/2Z-orbifold singularity, see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='2 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' It is relatively straightfor- ward to establish the equivalence between their characterization and the normal forms provided by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS 19 As in [37], an immediate consequence of the normal forms in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='4 is that every persistent nilpotent singular point can immediately be turned into elementary ones by means of a single blow-up of weight 2, see [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In closing this section, let us point out the family of vector fields described in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='4 is a genuine extension of the Sancho-Sanz family, albeit one naturally obtained by following their construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Indeed, for persistent nilpotent singular points, the multiplicity of the foliation along the corresponding (formal) separatrix is fully invariant under blow ups whose centers are contained in singular sets, cf, Formula 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In the Sancho-Sanz family, all multiplicities are equal to 2 so that for n ≥ 3, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='4 yields examples that cannot be turned in Sancho-Sanz examples by means of successive blow ups (and conversely).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' For example, vector fields in the family Xλ = (y − λz) ∂ ∂x + zx ∂ ∂y + z3 ∂ ∂z , with λ ̸= 0, yield foliations Fλ with persistent nilpotent singularities arising from a (strictly) formal separatrix Sλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The multiplicity of Fλ along Sλ being equal to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The desingularization theorem of McQuillan-Panazzolo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The purpose of this para- graph is to explain in detail the desingularization theorem proved in [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' As previously men- tioned, McQuillan and Panazzolo work from the start in the category of weighted blow-ups, thus not limiting themselves to stardard ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Unlike standard blow-ups, that keep the smooth nature of the space, weighted blow-ups lead to singular ambient spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Yet, the singularities in question are of orbifold-type and hence of a rather simple nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Whereas singular, it should be pointed, that the ambient space obtained after a sequence of finitely many weighted blow ups still is birationally equivalent to the initial one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, foliations can be transformed without any restrictions under weighted blow ups to yield new birational models for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Keeping in mind the issues pointed out above, let us summarize the contents of [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The paper [37] is essentially divided into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Its first part is devoted to prove that the algorithm of [39] - leading to a resolution of singularities by means of a sequence of weighted blow-ups for real analytic foliations on (R3, 0) - applies equally well in the general case of holomorphic foliations on (C3, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The algorithm in question thus provides a birational model for the foliation on a space possessing orbifold-type singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Note that, since we are dealing with (singular) foliations on spaces with singular points of orbifold type, a word is needed about the meaning of “elementary singular points”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In this regard, the singular point of the foliation is said to be elementary if it is given by an elementary singular point in a orbifold coordinate for the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, there are an open set U ⊂ C3 and finitely ramified map from U to a neighborhood of the orbifold singular point such that when the foliation is pulled-back to the open set U ⊂ C3 only elementary singular points are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In the second part of [37], the authors consider the problem of resolving the singular points of the ambient space while keeping the singular points of the foliation elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' They prove that a resolution for such singularities exists except when the singular point correspond to a Z/2Z-orbifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In other words, they have shown that, given a foliation F, it is always possible to obtain a birational model for F possessing only Z/2Z-orbifold singular points and where all the singular points of the foliation in question are elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In turn, the singularities asso- ciated with Z/2Z-orbifolds that appear in the end of the previous construction can be turned into elementary singularities by means of a single blow-up of weight 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' These singularities asso- ciated with Z/2Z-orbifolds actually correspond to the previously described persistent nilpotent singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The result in [37] can thus be stated as follows: 20 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REBELO AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REIS Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [37] Let F be a singular holomorphic foliation on (C3, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' There is a sequence of weighted blow-ups (5) F0 π1 ←− F1 π2 ←− · · · πl ←− Fl satisfying the following conditions: (i) The center of each weighted-blow up is strictly invariant with respect to the quasi- homogeneous filtration in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (ii) The ambient space is an analytic space of dimension 3 whose singular points are Z/2Z- orbifold type and the total blow-up map π1 ◦ · · · ◦ πl is birational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (iii) The singular points of Fl are elementary in orbifold coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let us close this paragraph with a comment concerning item (i) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Note that this item is not emphasized in [37] though it is a characteristic property of Panazzolo’s algorithm in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Whereas, as far as foliations are concerned, this is a minor issue - as it would also be the case of blow ups centered away from the singular locus (whether or not the blow ups are weighted) - the issue becomes relevant when our main interest lies in vector fields, rather than foliations, see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Resolution following [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In [49], we also establish the existence of a birational model for F where all singularities of F are elementary except for finitely many ones that can be turned into elementary singular points by means of a single blow-up of weight 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' To be more precise, our resolution result for foliations can be stated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [49] Let F denote a singular holomorphic foliation defined on a neighborhood of (0, 0, 0) ∈ C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Then there exists a finite sequence of blow-up maps along with transformed foliations (6) F = F0 π1 ←− F1 π2 ←− · · · πl ←− Fn satisfying all of the following conditions: (1) The center of the blow-up map πi is (smooth and) contained in the singular set of Fi−1, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (2) The singularities of Fn are either elementary or persistently nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (3) The number of persistently nilpotent singularities of Fn is finite and each of them can be turned into elementary singular points by performing a single weighted blow-up of weight 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The proof of this theorem has essentially two main ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The first one concerns a (personal) comment by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Cano claiming that “if a foliation cannot be resolved by standard blow ups, then there must exist a formal separatrix giving rise to a sequence of infinitely near singular points that never becomes elementary”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This assertion harks back to his earlier works on resolutions of 1-dimensional foliations [6] and some important results in this direction can also be found in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' To provide a complete proof of Cano’s assertion was therefore a crucial point in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='6 and the corresponding result is the content of Proposition 4 in [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Interestingly enough, the argument provided in [49] is rather different from the one envisaged by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Cano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' With Proposition 4 of [49] in place, the main idea to derive Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='6 is to argue from the notion of multiplicity of a foliation along a separatrix, as defined in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The sequence formed by a separatrix and its transforms is decreasing so that it stabilizes after finitely steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' When the sequence becomes stable, the order of the singular point of the foliation must be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Thus either the singularity has become elementary or we can resort to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='4 to characterize it as a persistent nilpotent singularity, which is necessarily isolated among other possible persistent nilpotent singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Therefore this yields a local uniformization theorem GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS 21 in the sense of Zariski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' At this point, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Piltant “gluing theorem” [40] allows one to conclude Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Recalling that persistent nilpotent singular points are in correspondence with Z/2Z-orbifold type singular points, the differences between Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='5 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='6 are down to the way the corresponding birational models are constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Unlike Panazzolo [39], our proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='6 does not provide any effective algorithm to resolve singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In some problems, however, it might simplify discussions/arguments by sticking to a single type of blow up, the standard one, provided that the problem in question requires only a theorem asserting the existence of a resolution, as opposed to an effective manner to obtain the resolution in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' A final comment on transforming vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In close this section, let us point out a virtue of standard blow ups, as used as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='6, that is also present in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='5 thanks to item (i) in the corresponding statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This concerns vector fields as opposed to 1-dimensional foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In fact, it is not a foliation but rather some holomorphic vector field that is the object of primary interestin many problems and applications of singularity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Examples of this situation are provided in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='1 and throughout Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Naturally a vector field X gives rise to an 1-dimensional foliation F of which a birational model whose all singular points are elementary may be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Nonetheless, if the vector field X is the object of primary interest, then the transforms of X have to be considered as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' At this point, the difference between vector fields and 1-dimensional foliations is summarized by the following self-evident statement: the transform of a 1-dimensional holomorphic foliation under a rational map is another 1-dimensional holomorphic foliation, however, the transform of a holomorphic vector field under a rational map is, in general, a meromorphic vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' When applying resolution theorems for 1-dimensional foliations to the study of vector fields it is therefore relevant to seek to retain the “good” analytic properties of them as much as possible (again concrete examples are provided in Sections 5 and 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' For example, if we start with a holomorphic vector field X, we might hope that its transform at the end of a resolution procedure still is a holomorphic vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In this context, we have: Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The transform of a holomorphic vector field under a resolution procedure as in Theo- rem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='5 or in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='6 still is a holomorphic vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The claim is clear in the case of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='6 as it is a basic fact that the blow-up of a holomorphic vector field centered at its singular locus is again holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In fact, for the blown-up vector field to be holomorphic again it suffices the center of the blow-up to be invariant by the initial vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In the case of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='5 this is not immediate as a weighted blow-up may turn a holomor- phic vector field into a meromorphic one even if its center is contained in the singular set of the initial vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This is where the condition of having centers that are called strictly invariant with respect to the quasi-homogeneous filtration in question, as used in [39] and reproduced in the first part of [37], comes into play (see item (i) in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This condition, if slightly technical, ensures that the transform of holomorphic vector fields remains holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Invariant analytic sets The problem of existence of invariant manifolds has always been a central theme in the theory of dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Among the many reasons for this, there is the simple fact that these invariant manifolds usually provide reductions on the dimension of the corresponding phase-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' For example, in the general theory of hyperbolic systems, the so-called stable manifolds are examples of invariant manifolds and, in fact, their existence form a cornerstone 22 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REBELO AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REIS of the hyperbolic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The existence of stable manifolds for hyperbolic singular points is a consequence of the general theory and ensured by the well-known Stable Manifold Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' However, whether or not “stable”, invariant manifolds may fail to exist if the singular point is no longer hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The simplest example is provided by the vector field X = y ∂ ∂x − x ∂ ∂y all of whose integral curves are circles around the origin of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Clearly there is no invariant manifold in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The general problem of existence of invariant manifolds may also be considered in the context of holomorphic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In this case, we look for invariant complex-analytic objects, which is a much stronger regularity condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' We allow, however, these objects to be singular in the sense of analytic sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In other words, we look for invariant varieties, as opposed to actual manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In the sequel, the word “manifold” will be saved for smooth objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' As mentioned in section 2, Briot and Bouquet were the first to consider the problem of existence of separatrices for holomorphic vector fields defined on a neighborhood of the origin of C2 in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' However, they were not able to establish the existence of separatrices for all holomorphic vector fields on (C2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This question was settled only much later by Camacho and Sad in their remarkable paper [5] where the following is proved: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [5] Let F be a singular holomorphic foliation defined on a neighborhood of the origin of C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Then there exists an analytic invariant curve passing through (0, 0) and invariant by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='1 is well worth a few additional comments, namely: It is somehow surprising that separatrices for holomorphic vector fields on (C2, 0) always exist despite the much stronger regularity condition for the invariant curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' For example, for the holomorphic vector field y ∂ ∂x − x ∂ ∂y defined on (C2, 0), the separatrices are given by the two complex lines y = ±ix and hence are totally contained in the non-real part of C2 (bar the singular point itself).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' However, as mentioned, we do not require the separatrices to be smooth invariant curves otherwise no general existence statement would hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Indeed, as a simple example, consider the holomorphic vector field 2y∂/∂x + x3∂/∂y on (C2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Since this vector field admits f(x, y) = x3 − y2 as first integral, it immediately follows that the only separatrix of X is the cusp of equation {x3 − y2 = 0}, which is clearly not smooth at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Fortunately, allowing separatrices to be singular is not a problem, since Hironaka’s theorem can always be used to desingularize them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Also it is important to emphasize that Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='1 applies only to foliations defined on smooth ambients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In fact, if germs of foliations defined on singular surfaces are considered, then separatrices may fail to exist as shown by Camacho in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The existence of separatrices is, however, no longer a general phenomenon in dimension 3, regardless of the dimension of the foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' As already said, a first example of codimension 1 foliation on (C3, 0) without separatrices was provided by Jouanolou in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Jouanolou’s ex- ample essentially hinging from the core dynamics of (dicritical) foliations on (C3, 0), the same idea enables us to construct plenty of additional examples of codimension 1 foliations without separatrices (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='2 or the summary below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Concerning 1-dimensional foliations, examples of foliations without separatrices in dimen- sion 3 were found by Gomez-Mont and Luengo, [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Their work will be discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' For the time being, we will focus on the problem of invariant manifolds for codimension 1 foli- ations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS 23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Separatrices for codimension 1 foliations induced by pairs of commuting vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let us begin by recalling/summarizing the discussion in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='2 where it was shown how Jouanolou’s method can be used to produce many examples of codimension 1 foliations without separatrix on (C3, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (i) Every homogeneous polynomial vector field X on C3 that is not a multiple of the Radial vector field induces a foliation on CP2 corresponding to the so-called core foliation associated with X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Conversely, given a foliation on CP2, there exists a homogeneous polynomial vector field on C3 whose core foliation is the given one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (ii) Let X be a homogeneous vector fields distinct from a multiple of the Radial vector field R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' There follows from the Euler relation that X and R generates a Lie algebra isomor- phic to the Lie algebra of the affine group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, the distribution generated by X and R can be integrated to yield a dicritical codimension 1 foliation D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Furthermore, the core foliation associated with D coincides with the core foliation associated with X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (iii) Finally, for every fixed degree, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='9 ensures the existence of a (non-empty) open set of foliations on CP2 such that all leaves of each foliation F in this set are dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, no foliation in this set admits algebraic invariant curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The codimension 1 foliations generated by R and by the homogeneous vector field having one such foliation as core foliation has no seraparatrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In view of what precedes, it is natural to wonder if all example of codimension 1 foliations without separatrices are among the dicritical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In ambient spaces of dimension 3 this, in fact, holds as proved by Cano and Cerveau in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Their result can be stated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [7] Let D be a germ of a holomorphic singular codimension 1 foliation on (C3, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' If D is not dicritical, then it admits a separatrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The proof of the previous result relies heavily on a resolution theorem for non-dicritical codimension 1 foliations obtained by the authors in the same paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Note, however, that the non-dicritical assumption, implies that the transforms of the initial codimension 1 foliation leave every irreducible component of the exceptional divisor invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In other words, away from singular points, every irreducible component of the exceptional divisor is a leaf of the foliations in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This rules out the existence of meaningful core dynamics and making the problem very much comparable to the 2-dimensional situation handled by Camacho and Sad in [5] which has a more geometric nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In a different direction, experts including F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Cano, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Cerveau, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Stolovitch have since long wondered what would be the “correct generalization” of Camacho-Sad theorem for (C3, 0), already at level of codimension 1 foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, the idea that a codimension 1 foliation spanned by a pair of commuting vector fields (not everywhere parallel) might necessarily admit separatrices was advanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The question is settled by the theorem below which confirms their intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [44] Consider holomorphic vector fields X, Y defined on a neighborhood of the origin of C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Suppose that they commute and are linearly independent at generic points (so that they span a codimension 1 foliation denoted by D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Then D possesses a separatrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The remainder of this paragraph is devoted to single out a few issues involved in the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This illustrates several points made in the preceding sections, including the usefulness of “taming” core dynamics (and how “symmetries” may be exploited to this effect) and the role of resolutions theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Concerning the latter, the argument will also highlight the the importance of having actual vector fields - rather than mere foliations - being “nicely” transformed during the resolution procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 24 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REBELO AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REIS The first ingredient in the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='3 is therefore a general resolution of singularities theorem for codimension 1 foliations in dimension 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Compared to Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='2, the main result in [7] is arguably a theorem of reduction of singularities for the foliations in question under the additional condition that the foliation should be non-dicritical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Fortunately, Cano has obtained in [8] a general resolution theorem for codimension 1 foliations on (C3, 0) which applies equally well to dicritical foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Armed with Cano’s theorem [8], we see that the basic obstacle for the existence of separa- trices lies in the core dynamics by means of the phenomenon already pointed out in Jouanolou examples, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The central point in the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='3 is therefore to “tame” the core dynamics arising from dicritical divisors the resolution procedure applied to D will have “plenty of algebraic curves”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In the sequel, we shall indicate some simple ideas used to show that the mentioned core dynamics cannot be “too wild”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let us consider the simplest case where we want to blow-up the origin (a degenerate singular point of D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The first lemma shows that at least one between the vector fields X and Y have to induce a foliation on the resulting exceptional divisor, unless we have a truly very special situation that is essentially “linear” (and hence easy to handle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Recalling that D is spanned by the commuting vector fields X and Y , let FX (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' FY ) denote the 1-dimensional singular foliation associated with X (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Assume that the first jet of both X and Y at the origin are equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Then none of the foliations FX or FY is dicritical for the blow-up π of C3 at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Denote by � X and �Y the blow-ups of X and Y at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Similarly, �Fx and �FY will stand for the blow ups of the foliations FX and FY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Since the vector fields X and Y have zero linear part at the origin, there follows that both � X and �Y vanish identically over the exceptional divisor π−1(0) ≃ CP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Now assume that, say, X is dicritical for π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Then the leaf of �FX is regular and transverse to π−1(0) at generic points of π−1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Therefore, around one such point, we can choose local coordinates (u, v, w) such that {u = 0} ⊂ π−1(0) and where of �FX is represented by the (regular) vector field ∂/∂u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular the blow-up � X takes on the form f(u, v, w)∂/∂u where f is a holomorphic function (divisible by u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In these coordinates, let the blow-up �Y be given by �Y = f1∂/∂u+f2∂/∂v +f3∂/∂w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Since [ � X, �Y ] = 0, there follows that f2 and f3 do not depend on the variable u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' However, these functions must vanish identically for u = 0 since �Y vanishes identically over π−1(0) (locally given by {u = 0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Thus they must vanish identically over an open set and this contradicts the fact that X and Y span a codimension 1 foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The argument above shows the importance of transforming vector fields, as opposed to foliations, in certain cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In fact, the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='4 hinges from the fact that the transform of the vector field Y vanishes identically over the exceptional divisor π−1(0) something that does not make sense for a foliation since the singular set of the latter has codimension at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Along similar lines, to ensure that the transformed vector field �Y vanishes identically over π−1(0), the fact that the origin (center of the blow up) is contained in the singular set of FX (or more generally, the center of the blow up is invariant under the foliation) was implicitly used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This is in line with the discussion in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' It is often important that the transformed vector field retains its holomorphic character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In addition, in quite a few cases, it is also important that the zero-divisor of the transformed vector field contains all components of the exceptional divisor arising from the resolution procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Plenty of additional examples of this issue can be found in the theory of semicomplete vector fields, see for example [13], [18], or [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS 25 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='4 shows that both FX or FY must induce a foliation on CP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' If the foliations induced are different, then it follows from the discussion in Section 3 that π−1(0) is invariant by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Hence we can assume that they do coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Recalling that the order of a vector field at a singular point p is nothing but the degree of the first non-zero homogeneous component of its Taylor series based at the point in question, the preceding implies: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The orders at the origin of X and Y can be assumed to be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Assume that X and Y have the same order at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Because they induce the same foliation on CP2, they will differ by a multiple of the radial vector field (up to multiplying, say X, by a non-zero constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Hence, by considering Z = X − Y , there follows that the foliation D is still spanned by X and Z and the we still have [X, Z] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This is however impossible since Z is clearly dicritical so that a contradiction with Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='4 arises at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' □ Denote by XH (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Y H) the first non-zero homogeneous component of X (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Y ) at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Owing to the above lemma, we can assume that the degree of Y H is strictly greater than the degree of XH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The preceding implies that the core dynamics of either XH, Y H coincides with the core dynamics of the dicritical foliation D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Now the following proposition provides some serious control on the core dynamics in question and, along with its analogue for blow ups centered at curves, constitutes a fundamental starting point of the discussion conducted in [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The vector field XH admits a non-constant meromorphic/holomorphic first integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Owing again to the discussion in Section 3, the dicritical nature of D ensures the existence of holomorphic functions f and g such that (7) fX + gY = Z , with Z being a holomorphic vector field whose first non-zero homogeneous component at the origin is a multiple of the radial vector field R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Denoting by f H, gH the first non-zero homoge- neous components of f, g, there follows from the preceding that f HXH and gHY H must have the same degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Furthermore, we have a homogeneous equation (8) f HXH + gHY H = hHR where hH is a homogeneous polynomial - possibly identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In the sequel we assume that hH does not vanish identically since it is easy to adapt the discussion below to cover this case as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Because X, Y commute, so do XH, Y H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Thus we have [XH, Y H] = � XH, hH gH R − f H gH XH � = � XH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' �hH gH �� R − hH gH [R, XH] − � XH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' �f H gH �� XH = � XH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' �hH gH �� R − � (d − 1)hH gH − XH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' �f H gH �� XH = 0 where d stands for the degree of XH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular � XH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' �hH gH �� R = � (d − 1)hH gH − XH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' �f H gH �� XH .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 26 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REBELO AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REIS The expression between brackets on the left hand side (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' the expression multiplying R) must vanish identically for otherwise XH would be a multiple of the Radial vector field R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' It then follows that XH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' �hH gH � = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In other words, hH/gH is a meromorphic (possibly holomorphic) first integral for XH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' It only remains to prove that hH/gH is not constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' However, if this function is constant (different from zero since hK does not vanish identically), then we can assume hH/gH = 1 without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Hence dividing (8) by gH, it would follow f H gH XH + Y H = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This last equation is however impossible since Y H has degree at least 2 and the expression f HXH/gH is homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Therefore hH/gH cannot be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Since the argument is symmetric in the vector fields X, Y , the last assertion completes our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' □ The key to prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='3 is to observe that the core dynamics of dicritical components of a foliation like D must leave invariant certain algebraic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Clearly Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='7 along with some refinements play a role in this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' However, it is also clear that the discussion leading to Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='7 depends heavily on the vanishing assumption for the first jet of X, Y at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This issue requires to consider separately some special situations that are referred to as “linear foliations” in the sense that there is a non-zero first jet involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Not surprisingly, “linear foliations” can be dealt with through rather direct methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Finally, as it is inevitable in dimension 3, every desingularization procedure requires two types of blow-ups: beyond blow-ups centered at points, blow-ups centered at curves are needed as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, another basic ingredient in the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='3 will be analogues, both in “linear” and “non-linear” settings, of the previous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This issue has already appeared in Section 4 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' the discussion about Equation 4) and can easily lead to misunderstandings so that it seems convenient to close this paragraph by carefully explaining the appropriate formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The following example was pointed out to us by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Cerveau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' It helps to explain the notion of “zero first jet” in the case of blow-ups centered at smooth curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The example also high- lights difficulties related to the existence of first integrals (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='7 and a few other intermediary results used in [44] and not explicitly mentioned here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Consider the pair of vector fields X, Y given by X = zy ∂ ∂y + z2 ∂ ∂z and Y = x2 ∂ ∂x + axy ∂ ∂y which are quadratic at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Note that these two vector fields commute so that they span a codimension 1 foliation denoted by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The axis {y = z = 0} is invariant by both X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This axis is also contained in the singular set of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let us then consider the blow-up of C3 centered at the axis {y = z = 0} along with the corresponding transforms of D, X, and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' It is immediate to check D is dicritical for the blow up in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Similarly the foliation FX associated with X is also dicritical for this blow up which might lead to some confusion with Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The explanation for this example lies in the fact that the vector field Y is regular (non-zero) at generic points of the axis {y = z = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Similarly, its transform under the previous blow up is regular at generic points of the exceptional divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In other words, this case must be GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS 27 considered as a “linear one” and the order of Y with respect to this blow up must, indeed, be equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' An adequate definition of the order of a vector field with respect to the center of a blow up is included below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let us then provide an accurate definition of order of a vector field when a curve, as opposed to a single point, is blown-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' To explain the idea, consider first a holomorphic vector field X with a singular point at the origin along with the corresponding Taylor series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The order of X at the origin is said to be the degree of the first non-zero homogeneous component of its Taylor expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This can also be viewed as the integer d for which the limit lim λ→0 1 λd−1 Γ∗ λX yields a (non identically zero) holomorphic vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Here Γ∗ λX stands for the pull-back of X by the homothety Γλ : (x, y, z) �→ (λx, λy, λz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Note that the limit above corresponds to the first non-zero homogeneous component of the Taylor’s expansion of X at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Next, assume now that C = {y = z = 0} is contained in the singular set of X so that the blow-up centered along this curve of singular points will be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The order of X with respect to C is defined as the integer d for which lim λ→0 1 λd−1 Λ∗ λX is a (non identically zero) holomorphic vector field, where Λ∗ λX denotes the pull-back of X by the homothety Λλ : (x, y, z) �→ (x, λy, λz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The limit above, for the appropriate choice of d, is said to be the first non-zero homogeneous component of X with respect to the variables x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In general, the cases in [44] that are called linear are those cases in which the vector field has order 1 or zero, with respect to the center of the blow up in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, with the above definition, it can immediately be checked that the vector field Y of Example 1 has order zero with respect to C = {y = z = 0}, although its order at the origin is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Separatrices for foliations of dimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This paragraph is devoted to discussing in detail the problem about existence of separatrices for 1-dimensional foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Contrasting with the case of codimension 1 foliations, it will soon be seen that the influence of core dynamics in the existence of these separatrices is rather limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In fact, the existence of separatrices for foliations of dimension 1 is an phenomenon having, in a suitable sense, a very local nature: it essentially hinges from two basic ingredients, namely: The analysis of simple singularities which is basically conducted by direct methods involving normal forms and divergent series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Geometric considerations involving the relative positions of the simple singularities in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In this regard, and provided that a convenient resolution of singularities theorem is available, the problem somehow retains the same nature regardless of the dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' More precisely, the difficulties arising from increasing the dimension stem either from the evident fact that simple singularities are not always easy to describe (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' saddle-nodes of high codimension) and from the fact that the number of possible arrangements of their relative positions increase as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' As previously said, after Camacho-Sad theorem in [5] establishing the existence of separatrices for every foliation on (C2, 0), Gomez-Mont and Luengo found a foliation on (C3, 0) that admits no separatrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let us begin by providing an outline of their construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' On Gomez-Mont and Luengo counterexample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Their example of foliation without separatrix on (C3, 0) relies on two simple remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Consider a foliation F on (C3, 0) given by a holomorphic vector field satisfying the following conditions 28 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REBELO AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REIS (1) The origin (0, 0, 0) ∈ C3 is an isolated singularity of X (2) J1X(0, 0, 0) = 0 but J2X(0, 0, 0) ̸= 0, where JkX(0, 0, 0) stands for the jet of order k of X at the origin (k = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (3) The quadratic part X2 of X at (0, 0, 0) is a vector field whose singular set has codimen- sion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Also X2 is not a multiple of the Radial vector field x∂/∂x + y∂/∂y + z∂/∂z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Assume that F has a separatrix C and consider the blow-up �F of F centered at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Denote by π the blow-up map so that �F = π∗F and let π−1(0) denote the exceptional divisor isomorphic to CP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Since X2 is not a multiple of the Radial vector field, there follows that π−1(0) is invariant by � F so that the restriction of �F to π−1(0) can be seen as a foliation of degree 2 on CP2 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' item (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Because π−1(0) is invariant by �F, the transform π−1(C) of the separatrix C can only intersect π−1(0) at singular points of �F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Furthermore, all of these singular points lie in π−1(0) since X has an isolated singularity at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In other words, π−1(C) must be a separatrix (not contained in π−1(0)) for one of the singular points of �F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Now, the second ingredient is as follows: as a foliation of degree 2 on CP2, the restriction �F|π−1(0) of �F to π−1(0) has at most (and generically) 7 singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Since it is hard to control the position of 7 points in CP2, the authors of [15] started from a foliation satisfying the following conditions: (A) The foliation of degree 2 has only 3 singular points (we can think of the foliation as obtained by letting some of the 7 singular points of a generic quadratic foliation to “collide in groups”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Naturally the position of 3 points in CP2 can easily be controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (B) Each of the 3 singular points will have an eigenvalue equal to zero in the direction transverse to π−1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The 3 singular points are therefore saddle-node singularities (in dimension 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (C) Furthermore, the authors arrange for the saddle-node singularities to have two equal (and non-zero) eigenvalues tangent to π−1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In other words, the singular points in question are (codimension 1) resonant saddle-nodes with weak direction transverse to π−1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (D) As is well known, it is easy to produce examples of codimension 1 saddle-nodes all of whose separatrices are included in an invariant (2-dimensional) plane tangent to the directions of the non-zero eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The remainder of the proof in [15] consists of showing that it is, indeed, possible to prescribe a quadratic X2 and a cubic X3 homogeneous components for the vector field X so as to satisfy all of the preceding conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In this respect, note that conditions (A), (B), and (C) depend only on the quadratic part X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The role played by the appropriately chosen cubic parte X3 can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' it ensures that each of the singular points of �F are isolated, hence coinciding with the corresponding singular points of �F|π−1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Here the reader may note that the homoge- neous foliation associated with X2 has singularities all along the fibers of �C3 → �π−1(0) sitting over the singular points of �F|π−1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' A higher order perturbation of X2 is thus needed to provide isolated singular points for the blown-up foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' having ensured the singular points are isolated, the cubic part X3 of X also takes care of condition (D) As mentioned, the verification that all these conditions are compatible is conducted in [15] with the assistance of suitable software to deal with formal computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS 29 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Vector fields and 2-dimensional Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In [44], codimension 1 foliations spanned by pairs of commuting vector fields were considered and it was shown that this condition im- poses strong constraints on the core dynamics of dicritical components of the codimension 1 foliation in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, these constraints have proved to be strong enough to yield the existence of separatrices for the foliation in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In view of the preceding, it was natural to wonder if the 1-dimensional foliations arising from the vector fields in question would have separatrices themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' While the answer turned out to be affirmative, the assumption of having two commuting vector fields can be weakened to encompass also the case of pairs of vector fields generating the Lie algebra of the affine group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In fact, the following theorem was proved in [47]: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [47] Let X and Y be two holomorphic vector fields defined on a neighborhood U of (0, 0, 0) ∈ C3 which are not linearly dependent on all of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Suppose that X and Y vanish at the origin and that one of the following conditions holds: [X, Y ] = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [X, Y ] = c Y , for a certain c ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Then there exists a germ of analytic curve C ⊂ C3 passing through the origin and simultaneously invariant under X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The theorem above deserves a few additional comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' First, the fact that Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='8 applies to pair of vector fields generating the Lie algebra of the affine group is in stark contrast with the analogous problem for codimension 1 foliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Indeed, every homogeneous vector field of degree at least 2 together with the radial vector field generate the Lie algebra of the affine group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, Jouanolou’s and similar examples of codimension 1 foliation without separatrices arise from pairs of vector fields generating the Lie algebra of the affine group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Whereas theorems asserting the existence of separatrices for foliations of dimension 1 holds interest in their own right, they also have non-trivial applications in the general problem of understanding globally defined holomorphic vector fields on compact com- plex manifolds, see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, the paper [47] also includes a non-trivial applications of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='8 in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Finally, a relatively minor but yet subtle issue that is worth pointing out is that Theo- rem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='8 claims that X and Y possess a common invariant curve without asserting that the curve in question is invariant by the foliations associated with X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' To further clarify the issue, it is enough to think of the 2-dimensional vector field x∂/∂x: the axis {x = 0} is invariant by the vector field but does not constitute a separatrix for the associated foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In turn, it might be asked if the foliations associated with X and Y share an actual separatrix, possibly enlarging the notion of “separatrix” to include curves fully constituted by singular points of the corresponding foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, it is easy to check that the existence of “common separatrices” always holds when X is a homogeneous vector field and Y is the radial vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Indeed, in this case the leaves of FY are simply the radial lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Concerning FX, since it is not a multiple of the Radial vector field, it induces a 1-dimensional foliation on CP2 by means of the one- point blow-up of C3 at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The foliation in question possesses isolated singular points and it can easily be checked that the radial line naturally associated with any of these singular points is invariant by FX as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' We believe that the existence of a common separatrix for FX and FY in the general case can also be established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' To finish the section, let us provide an outline of the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 30 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REBELO AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REIS Sketch of Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Recall that the foliation associated with X (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Y ) is de- noted by FX (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' FY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let D denote the codimension 1 foliation spanned by X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' We have that codim (Sing (D)) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In other words, Sing (D) is of one of the following types: the union of a finite number of irreducible curves, a single point (the origin), or simply empty (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' D is regular).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Since Sing (D) is naturally invariant by X and by Y , the result immediately holds if dim (Sing (D)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Hence we can assume without loss of generality that Sing (D) has codimension at least 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In other words, either Sing (D) is reduced to the origin or it is, in fact, empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Since the singular set of D has codimension at least 3, Malgrange Theorem [29] implies that D possesses a non-constant holomorphic first integral f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let then S = f −1(0) so that S is an invariant surface for D, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' the irreducible components of S are separatrices for D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, S is invariant by both X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Next, note that S can be assumed to be irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Otherwise, the intersection of any two irreducible components of S yields a curve invariant under both X, Y and the conclusion holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The surface S can then be assumed either regular or having an isolated singularity at the origin (again if S contains a curve of singular points this curve must be invariant by X and Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' At this point, a couple of remarks are in order: In the case where S is smooth, each of the foliations FX and FY possesses separatrices owing to Camacho-Sad Theorem [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Still it remains to check that these foliations share a common separatrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' As previously mentioned, in the case of singular surfaces, there are examples of foliations without separatrix (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [4] or [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This phenomenon needs thus to be ruled out in the present case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In general, we proceed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Consider the restrictions of X and Y to S along with the corresponding tangency locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This tangency locus is not empty since both X and Y vanish at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Since the tangency locus Tang (X|S, Y |S) is invariant by both X and Y , the result immediately holds in the case where its dimension equals 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' So, we shall consider separately the case where Tang (X|S, Y |S) = {(0, 0, 0)} and the case where Tang (X|S, Y |S) = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Assuming that Tang (X|S, Y |S) is reduced to the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Then S is a surface with singular set of codimension at least 2 and equipped with two vector fields that are linearly independent away from this an analytic set of codimension 2 or greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This implies that tangent sheaf to S is locally trivial which, in turn, implies that S is smooth since S is a hypersurface in C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' However, being smooth, S is locally equivalent to C2 and the tangency locus of two vector fields cannot be reduced to a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The resulting contradiction rules out this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Assume now that Tang (X|S, Y |S) = S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' the restrictions to S of X and Y coincide up to a multiplicative function (defined on S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The existence of the desired common separatrix is then ensured in the case where S is smooth by Camacho-Sad theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' It only remains to consider the case where S has an isolated singular point at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The argument in this case relies on proving that the (1-dimensional) foliation induced on S by either X or Y possesses a non-constant holomorphic first integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The level curve of this first integral containing the origin then yields the desired separatrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Details can be found in [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Semicomplete vector fields, automorphism groups, and separatrices The object of this last section is a distinguished class of singularities of vector fields, namely the semicomplete (singularities of) vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Understanding this class of vector fields, both at global level and at level of germs, is a problem with interesting applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' As an example of application, we will see in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='2 that results on singularities of semicomplete vector fields yield insight in some problems about bounds for the dimension of automorphism group of GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS 31 compact complex manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Another motivation to study these vector fields and their singular points stems from the very fact that the semicomplete property is somehow akin to the Painlev´e property for differential equations, albeit the two notions are not equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' As a matter of fact, as it happens with Painlev´e property, semicomplete vector fields are also largely present - sometimes implicitly - in the literature of Mathematical Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The notion of semicomplete singularity was introduced in [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The idea begins with the definition of semicomplete vector fields on general open sets which is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [41] A holomorphic vector field X defined on an open set U of some complex manifold M is said to be semicomplete (on U) if for every p ∈ U there exists a connected domain Vp ⊂ C, with 0 ∈ Vp, and a map φp : Vp → U satisfying the following conditions: φp(0) = p φ′ p(T) = X(φp(T)), for every T ∈ Vp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' For every sequence {Ti} ⊂ Vp such that limi→∞ Ti = ˆT ∈ ∂Vp the sequence {φp(Ti)} escapes from every compact subset of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The third condition in Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='1 basically means that φp : Vp → U is a maximal solution of X in a sense similar to the notion of “maximal solutions” commonly used for real vector field and/or differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In this sense, the definition is equivalent to saying that a vector field is semicomplete if for every p ∈ U the integral curve φ satisfying φ(0) = p has a maximal domain of definition in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Closely connected to the notion of maximal domain of definition in C, we can think of a local integral curve for a vector field X and then extending it over paths which is always possible as long as we stay in the domain of definition of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The vector field X is then semicomplete if these extensions do not give rise to any monodromy and hence can be merged together in a single (univalued) solution for X which is naturally defined on a maximal domain in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Though global in essence, the above definition has also a local character that is singled out by the following assertion: if a vector field X is semicomplete on U, then the restriction of X to every subset V of U is semicomplete as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Thus the notion of germ of semicomplete vector field, and hence of semicomplete singularity, makes sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Furthermore, even at level of germs, the condition of being semicomplete is far from trivial and, in fact, imposes strong constraints on the singular points of vector fields as pointed out in [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' As a matter of fact, since its introduction, semicomplete singularities have proved time and again that they capture almost all of the “intrinsic nature” of germs of vector fields admitting actual global realizations as complete vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Germs of holomorphic semicomplete vector fields on (C2, 0) were classified by Ghys and Rebelo in the papers [41] and [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, all these vector fields admit a non-constant holomorphic/meromorphic first integral so that the dynamics associated with them is rather simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' After this brief introduction to semicomplete vector fields, the remainder of the section will focus on two fundamental questions related to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The first question was somehow motivated by the results of Ghys and Rebelo in dimension 2 and asks the extent to which the condition of being semicomplete may tame the core dynamics of the corresponding foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In other words, we ask: Are there semicomplete vector fields exhibiting a genuinely complicated core dynamics?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The second question was raised by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Ghys long ago and, roughly speaking, involves deciding “how degenerate” can semicomplete singular points be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' A prototypical question along these lines concerns semicomplete vector fields with isolated singular points and can be formulated as follows: 32 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REBELO AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REIS Is it true that the second jet of a semicomplete vector field at an isolated singular point is necessarily different from zero?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This question is affirmatively answered in dimension 2 in the mentioned works by Ghys and Rebelo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' It remains open in higher dimension, though a number of partial results are available in dimension 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Whereas the interest in “taming” the core dynamics associated to singularities of vector fields has already been emphasized, let us also point out that the general question raised by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Ghys has applications to problems about bounds for the dimension of automorphism group of compact complex manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This issue will further be discussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' For the time being, we will focus on the dynamics associated with semicomplete singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Semicomplete vector fields with complicated dynamics - Guillot’s work [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' As previously mentioned, singularities of semicomplete vector fields have very simple dynamics in complex dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In fact, even the global behavior of semicomplete vector fields is amenable to detailed analysis, see [19], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' However, this is no longer the case in dimension 3 as follows from Guillot’s deep work on Halphen vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This paragraph is basically devoted to summarizing the main dynamical issues appearing in semicomplete Halphen vector fields while referring to [17] for a more comprehensive discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Halphen vector fields were first considered by Halphen himself [20], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Apart from his contribution, let us make clear that all remaining results in this paragraph are due to Guillot and can be found in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Up to linear equivalence, Halphen vector fields form a three parameters family of homogeneous polynomial vector fields of degree 2 on C3 explicitly described as X = � α1z2 1 + (1 − α1)(z1z2 + z1z3 − z2z3) � ∂ ∂z1 + (9) � α2z2 2 + (1 − α2)(z1z2 − z1z3 + z2z3) � ∂ ∂z2 + � α3z2 3 + (1 − α3)(−z1z2 + z1z3 + z2z3) � ∂ ∂z3 An alternate definition pointed out in [17] which already sheds some light in the intrinsic nature of these vector fields is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' A homogeneous polynomial vector field of degree 2 (a quadratic vector field for short) on C3 is Halphen if it satisfies the following relation (10) [C, X] = 2R , where C stands for a constant vector field and R is the Radial vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The normal form indicated in (9) is obtained as the solutions of Equation (10) for C = ∂/∂z1 + ∂/∂z2 + ∂/∂z3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Since both C and X are homogeneous, Euler relations imply that we also have [R, C] = −C and [R, X] = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In turn, these three relations together mean that the triplet {R, C, X} generates the Lie algebra of SL (2, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let FX, FR, and FC denote the 1-dimensional foliations associated to the vector fields X, R and C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Once again, let �C3 denote the blow-up of C3 centered at the origin with projection π : �C3 → C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The exceptional divisor π−1(0) is isomorphic to CP2 and the blow ups of FX, FR, and FC will respectively be denoted by �FX, �FR, and �FC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Similarly, � X, �R, and �C will stand for the blow ups of X, R, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Next, recall that, whenever two vector fields commute, then the flow of one of them will preserve the foliation associated with the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This simple remark hints at a basic property of Halphen vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Indeed, since X and C commute up to GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS 33 the Radial vector field, the flow of X “tends” to preserve the projection of the foliation arising from C along the orbits of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' To make this remark accurate, we first note that the space of orbits of R is naturally identified with the exceptional divisor π−1(0) ≃ CP2 though, on π−1(0), � X vanishes identically and �C has poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' However, the restrictions �FX|π−1(0) and �FC|π−1(0) to π−1(0) of the foliations �FX and �FC have a specific property of “mutual transversality” which is reminiscent from the previous observation on commuting vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This can be stated as follows: Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Two (singular) foliations F1 and F2 are said to be mutually transverse if they are (regular and) transverse away from an algebraic curve C which, in addition, is invariant by both F1 and F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, the curve C contains all singular points of F1 and of F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Keeping in mind that �FC|π−1(0) is nothing but a pencil of projective lines, the “mutual transversality” condition makes it easy to work out the structure of �FX|π−1(0) directly on π−1(0) ≃ CP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Namely, we have: Generically, �FX|π−1(0) leaves exactly 3 projective lines C1, C2 and C3 invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' These projective lines belong to the pencil �FC|π−1(0) and they intersect mutually at a radial singularity in π−1(0) (the base locus of the pencil) which is given in homogeneous coordinates by [1, 1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Also, the eigenvalues of �FX|π−1(0) at [1, 1, 1] are 1 and 1 (radial singularity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In fact, [1, 1, 1] is a radial singularity for the foliation in the 3-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In other words, the eigenvalue of �FX at [1, 1, 1] associated to the direction transverse to the exceptional divisor is again 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Away from the invariant projective lines C1, C2, and C3, the foliation �FX|π−1(0) is transverse to the remaining projective lines in the pencil �FC|π−1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Next, since X is homogeneous, the dynamics of the foliation �FX on �C3 can basically be recovered from the dynamics of the core foliation �FX|π−1(0) on π−1(0) ≃ CP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' We will return to this point later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In view of the preceding, let us first focus on the core foliation �FX|π−1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Note that both �FX|π−1(0) and the pencil �FC|π−1(0), the latter viewed as foliation, share the singular point [1, 1, 1] ∈ π−1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Consider the (2-dimensional) blow up of π−1(0) ≃ CP2 at [1, 1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The resulting surface is the Hirzebruch surface F1, the CP1-bundle over CP1 with a section of self-intersection −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' On F1, the foliation (pencil) �FC|π−1(0) becomes the standard fibration P : F1 → CP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In turn, the transform FX,F1 of the foliation �FX|π−1(0) on F1 is regular on a neighborhood of the −1-rational curve of F1 (identified with the exceptional divisor π−1(0) of the blow up of CP2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Also, there are 3 fibers of P that are invariant by FX,F1 and these fibers will still be denoted by C1, C2, and C3 by evident reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Away from these three fibers, FX,F1 is regular and transverse to the fibration induced by P on the open manifold F1 \\ {C1, C2, C3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The dynamics of � FX|π−1(0) can naturally be read off the dynamics of FX,F1 which, in turn, is essentially described by the holonomy representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In fact, the restriction of FX,F1 to the open surface (F1 \\ {C1, C2, C3}) is transverse to the restriction to (F1 \\ {C1, C2, C3}) of the fibration P : F1 → CP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Since the fibers of P are compact, Ehresmann’s observation ensures that the restriction of P to the leaves of FX,F1 yields a covering map from the leaf in question to CP1 \\ {z1, z2, z3}, where z1, z2, z3 are in natural correspondence with the invariant fibers C1, C2, C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The dynamics of FX,F1 is therefore essentially encoded in the holonomy representation, namely: the homomorphism ρ from the fundamental group of CP1 \\ {z1, z2, z3} 34 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REBELO AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REIS to the group of automorphisms of the fiber of P arising from parallel transport along leaves of FX,F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let π1(CP1 \\ {z1, z2, z3}) denote the fundamental group of CP1 \\ {z1, z2, z3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Since FX,F1 is holomorphic, the image of the holonomy representation ρ is contained in the group of holomor- phic diffeomorphisms of CP1 which can be identified with PSL (2, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The holonomy group Γ of FX,F1 is the image of π1(CP1 \\ {z1, z2, z3}) by ρ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Γ ⊂ PSL (2, C) is defined by Γ = ρ[π1(CP1 \\ {z1, z2, z3})].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Next, for each i = 1, 2, 3, let ξi ∈ PSL (2, C) be the holonomy map obtained by lifting a small loop around zi ∈ CP1 in the leaves of FX,F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The M¨oebious transformations ξ1, ξ2, ξ3 clearly generate the holonomy group Γ and satisfy the relation ξ1 ξ2 ξ3 = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' With the evident identifications, the dynamics of Γ on CP1 also accounts for the global dynamics of the foliation �FX|π−1(0) on CP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' All of the preceding considerations apply to every Halphen vector field in the family defined by (9), regardless of whether or not they are semicomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' To detect semicomplete Halphen vector fields in the family (9), we proceed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' First, notice that the singularities of �FX and of �FX|π−1(0) do coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Naturally there is the point [1, 1, 1] lying at the intersection of all the lines in the pencil �FC|π−1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Moreover, around [1, 1, 1], the foliation �FX is conjugate to the radial vector field in dimension 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' To describe the structure of the remaining singular points, for i = 1, 2, 3, let mi = (α1 +α2 + α3 − 2)/αi provided that αi ̸= 0, and set mi = ∞ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The remaining singular points of �FX are contained in the lines C1, C2, C3 and are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (1) If mi ̸= ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Then, aside from [1, 1, 1], �FX possesses exactly two singular points pi and qi in the line Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The eigenvalues of �FX at pi are −1, −1, −mi while at qi the eigenvalues are −1, −1, mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In both cases, the eigenvalues are ordered so that to the first eigenvalue corresponds to a direction transverse to the exceptional divisor, the second eigenvalue is associated with the direction of Ci and the third eigenvalue is associated with a direction transverse to Ci and contained in the exceptional divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (2) If mi = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Then, aside from [1, 1, 1], �FX possesses a unique singular point pi = qi in Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' At this singular point, the eigenvalues are −1, 0, −1 with the same ordering used in the above item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' When mi = ∞, the holonomy map ξi is a parabolic map in PSL (2, C) since �FX|π−1(0) has a (2-dimensional) saddle-node singularity at pi = qi with strong invariant manifold transverse to Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Next, we have: Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Assume that X is semicomplete and that mi ̸= ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Then mi is an integer (which can be assumed positive up to reversing the roles of pi and qi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Moreover the holonomy map ξi : CP(1) → CP(1) is periodic of period mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Again, up to renaming pi and qi, the singular point qi of �FX lies in the Siegel domain and the eigenvalues of the mentioned foliation at the singular point in question fulfill the conditions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' of Theorem 1 in [50] (or, equivalently, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='19 in [43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Furthermore, with the language of [50], [43], the eigenvalue that can be “separated” from the others by a straight line through 0 ∈ C is the first eigenvalue (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' the eigenvalue associated with direction transverse to the exceptional divisor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Consider then the separatrix S of �FX tangent to this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' It is immediate to check that the restriction of � X to S is given, in local coordinates, by −z2∂/∂z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Being X semicomplete, there follows that the local holonomy map of �F arising from a small loop in S encircling qi must agree with the identity, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Theorem 1 in [50] then ensures that � FX is linearizable around qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS 35 In particular, the foliation �FX|π−1(0) is also linearizable around qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' It follows that �FX|π−1(0) possesses a separatrix transverse to Ci and that the holonomy map arising from this separatrix is locally conjugate to a rotation of angle 2π/mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Because �FX|π−1(0) is transverse to a fibration, this local holonomy map is, in fact, the restriction of a global M¨oebius transformation ξi ∈ PSL (2, C) which, therefore, must verify ξmi i = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' □ As an immediate consequence, we have the following Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' If X is semicomplete, then the holonomy group Γ ⊂ PSL (2, C) describing the global dynamics of �FX|π−1(0) is given by Γ =< ξ1, ξ2, ξ3 : ξm1 1 = ξm2 2 = ξm3 3 = ξ1ξ2ξ3 = id > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In other words, Γ is a triangular group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='5, when mi = ∞, the condition ξ∞ i = id must be interpreted as simply saying that ξi is parabolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In the sequel, we shall also use the convention that 1/mi = 0 provided that mi = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In order to obtain semicomplete Halphen vector fields with complicate dynamics, we assume also that (11) m = 1 m1 + 1 m2 + 1 m3 < 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The effect of inequality (11) is just to rule out finitely many cases where the group Γ is “elementary”, either finite or conjugate to a subgroup of the affine group of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Assuming m1, m2, m3 fixed and as in (11), the resulting triangular group Γ satisfy all of the following conditions: The group Γ is unique (up to conjugation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Γ is discrete and non-elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Γ leaves a real projective line in CP1 invariant so that Γ is actually a non-elementary Fuchsian group (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Γ can also be viewed as a subgroup of PSL (2, R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The limit set Λ(Γ) of Γ coincides with the invariant circle S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, Γ acts properly discontinuously on each connected component of CP1 \\ Λ(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' As is well known, the dynamics of Γ on its limit set Λ(Γ) = S1 is very non-trivial: the dynamics has all orbits are dense and it is ergodic with respect to the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Also stationary measures are unique (and hard to understand in detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Clearly, these issues are directly reflected in the saturated of Λ(Γ) by the foliation �FX|π−1(0) whose dynamics is hence very non-trivial as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' It is also convenient to say a few words on the actual dynamics of �FX on �C3 rather than limiting ourselves to its core foliation �FX|π−1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' To describe this dynamics, we can follow essentially the same ideas used to describe the foliation �FX|π−1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Beginning with the pencil �FC|π−1(0), we define a family of surfaces in �C3 by considering the preimage of each line in �FC|π−1(0) by the canonical projection Π : �C3 → π−1(0) ≃ CP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' More precisely, for every projective line D in the pencil �FC|π−1(0), Π−1(D) is the line bundle over CP1 whose Chern class equals −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Alternatively, by adding a “section at infinity”, Π−1(D) can naturally be compactified into the Hirzebruch surface F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In other words, up to adding a “plane at infinity” to �C3, we obtain a family of F1 surfaces parameterized by the lines in the pencil �FC|π−1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Now, if we remove the three Hirzebruch surfaces sitting on the top of the lines in �FC|π−1(0) that are invariant under �FX|π−1(0), it is straightforward to conclude that �FX is transverse to the fibration by F1-surfaces over CP1 \\ {z1, z2, z3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Thus, once again we obtain a representation ρ from the 36 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REBELO AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REIS fundamental group of CP1 \\ {z1, z2, z3} in the group Aut (F1) of holomorphic diffeomorphisms of F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let Γ be the image of ρ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' the holonomy group of �FX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Clearly, ρ is generated by the maps Ξi obtained by lifting a small loop around zi, i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The maps Ξi can explicitly be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Fix a surface F1 equipped with coordinates (x, w) where x is projective coordinate on the projective line Π(F1) and w is an affine coordinate on the fibers of F1 that equals zero in the intersection with the exceptional divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Then we have (12) Ξi(x, w) = (ξi(x), � ξ′(x) w) c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Keeping in mind that the dynamics of �FX on �C3 and the dynamics of FX on C3 can be identified, what precedes can be summarized as follows (the slight abuse of language should not really lead to any misunderstanding): Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The dynamics of FX on C3 is essentially equivalent to the dynamics of the group Γ = ⟨Ξ1, Ξ2, Ξ3⟩ on F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, the (−1)-section of F1 is invariant by Γ and the restriction of the action of Γ to this section is nothing but the action of the triangular group Γ on CP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' By now, we have provided a description of the (rather non-trivial) dynamics of Halphen vector fields such that the quantities mi are integers satisfying the condition in (11) and the reader is referred to [17] for additional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' However, strictly speaking, we still do not know whether or not Halphen vector fields satisfying the conditions in question are, indeed, semicomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In fact, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='4 provides only necessary conditions for the vector field to be semicomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Hence, there remains the problem of checking that these conditions are also sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Curiously enough the fact that the corresponding Halphen vector fields are semicomplete is basically included in Halphen original papers [20], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Halphen begins his Note by pointing out that, if φ is a solution of a Halphen vector field, then so is (13) �φ = 1 (ct + d)2 φ �at + b ct + d � − c ct + d , for every a, b, c, d ∈ C with ad − bc ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' From this he concludes that all solutions can be described out of a single “known” solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' He then goes on to obtain a particular solutions by skillfully manipulating theta functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In this sense, the converse to Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='4 can be derived from his work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Yet, Guillot [17] provides a different proof of the semicomplete nature of Halphen vector fields satisfying the conditions in in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Guillot’s argument dispenses with the remarkable identities satisfied by theta functions and, perhaps more importantly, lends itself well to deep generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' We will close this paragraph by sketching this argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' First, it is convenient to recall the basic notions of translation, affine, and projective structures on Riemann surfaces since they play a role in the discussion below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let S be a Riemann surface along with a covering {(Bi, ϕi)} by local coordi- nates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The covering {(Bi, ϕi)} is said to define a translation structure (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' affine structure, projective structure) on S if and only if the changes of coordinates ϕi ◦ ϕ−1 j : ϕj(Bi ∩ Bj) → ϕi(Bi ∩ Bj) are restrictions of translations of C (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' affine maps, M¨oebius transformations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, if S is endowed with a nowhere zero holomorphic vector field X, then the covering whose local coordinates are the inverse maps of the (local) solutions of X endows S with a translation structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This simple remark will be useful below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Also, a translation structure (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' affine structure, projective structure) gives rise to a monodromy homomorphism ρ from the fundamental group of S to the group of translations of GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS 37 C (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' affine maps, M¨oebius transformations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Following [17], [19], denote by Sρ the covering space of S associated with the kernel of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' On Sρ, we can define a developing map Dρ : Sρ → C (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' C, CP1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In fact, Sρ is the smallest covering of S on which a developing map is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This developing map will be called the monodromy-developing map of the corresponding structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Naturally, all developing maps are well defined up to composition with an element of the corresponding group (translation, affine map, or M¨oebius transformations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The preceding offers us yet another equivalent way to define semicomplete vector fields on a Riemann surface, and thus in general since a vector field will be semicomplete if and only if its restriction to each leaf of its associated foliation is semicomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Namely, the vector field X on the Riemann surface S is semicomplete if and only if the monodromy-developing map of the corresponding translation structure is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' We are now ready to explain Guillot’s argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Sketch of Guillot’s proof that Halphen vector fields as in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='4 are semicomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' We might start by recalling that the vector fields R and C generate the Lie algebra of the affine group Aff (C, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Of course a similar remark applies to their blow ups �R and �C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Then we consider the Zariski open subset W of �C3 given as the complement of π−1(0) and of the 3 invariant Hirzebruch surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In the setting of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='6, W is a U-bundle over CP1 \\ {z1, z2, z3}, where U ⊂ F1 is the Zariski open set defined as the complement of the two rational sections of F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, U is in a natural correspondence with an orbit of Aff (C, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Finally, �FX is transverse to the fibers of the fibration W → CP1 \\ {z1, z2, z3} and admits a global holonomy group determined by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' It suffices to show that the restriction of � X to W is semicomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Guillot basic observation is that � X induces a natural projective structure on CP1 \\ {z1, z2, z3} viewed as the base of the U-bundle W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This deserves a few comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Small discs B ⊂ CP1\\{z1, z2, z3} can be identified with discs on the leaves L of �FX by means of the fiber bundle structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Next, each leaf L of the restriction of �FX to W is endowed with a translation structure induced by � X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' These transverse structures vary with the leaf but its underlining projective structure does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In fact, taking into account that a fiber U of the U-bundle W is identified with an orbit of Aff (C) and thus parameterized by the flows of R and of C, Equation (13) can be interpreted as an identity involving the flows of R, C, and X (or of their blow ups which amounts to the same).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' With this interpretation, it becomes clear that the time taken by � X to move between two fixed fibers U1 and U2 along leaves L and L′ are related by a M¨oebius transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Thus the covering of the base CP1 \\ {z1, z2, z3} obtained by taking the inverses of the local solutions of X, as above, over all possible leaves of � FX defines a projective structure on CP1 \\ {z1, z2, z3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Next, consider the monodromy-developing map Dρ for the projective structure CP1\\{z1, z2, z3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' It is straightforward to check that representatives for this developing-map can be obtained by simply considering the monodromy-developing maps associated with the translation structures induced by � X on the leaves of �FX (or more accurately of the restriction of �FX to W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In view of Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='8, there follows that X is semicomplete if and only if Dρ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Guillot’s then “compute” the projective structure in question by means of the Schwarzian operator so as to show that Dρ is essentially Schwarz triangular functions and the proof follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' □ Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In fairness, we should note that the material covered in this paragraph is essentially the first part of Guillot’s paper [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The content of [17] also includes realizing semicomplete Halphen vector fields as actual complete vector fields on complex manifolds as well as several important applications to the study of SL (2C) actions and homogeneous spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 38 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REBELO AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REIS Let us close this paragraph with a couple of questions about dynamics of semicomplete vector fields, the first one being kind of inevitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Are there semicomplete vector fields with complicated dynamics which gen- uinely different from the dynamics obtained by means of Halphen vector fields?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Another interesting question which may or may not have a saying in the above problem concerns geodesic flows on semisimple Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' These geodesic flows have already been considered in works by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Dumitrescu and by Elshafei-Ferreira-Reis, see [10], [12] and their references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Given a (semisimple) Lie group G and a left-invariant holomorphic metric on ⟨ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' ⟩ on G, the complex geodesic flow on G can be expressed by a quadratic vector field defined on the Lie algebra of G by means of the Euler-Arnold formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This yields a particular, yet large and with geometric nature, class of quadratic vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Referring to vector fields in this class as Euler-Arnold vector fields, their dynamics is definitely worth study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Thus we can formulate the following special case of the preceding question which, however, holds interest in its own: Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Are there semicomplete Euler-Arnold vector fields exhibiting complicated dy- namical behavior?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Local aspects of semicomplete vector fields and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Partly, the interest of semicomplete vector fields comes from the fact that they provide local obstructions for a germ of vector field be realized as singularity of a complete one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In the sequel, we will talk about germs of semicomplete vector fields or about semicomplete singularities as synonymous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' From the basic properties discussed at the beginning of this section, it follows that semicom- plete vector fields can be viewed as a “local counterpart” of complete ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In fact, a singularity that is not semicomplete cannot be realized by a complete vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, it cannot be realized by a globally defined holomorphic vector field on a compact manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The un- derstanding of semicomplete singularities is therefore useful to the description of holomorphic vector fields (globally) defined on compact manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' To better explain this issue, it is convenient to center the discussion around a rather concrete and well known question due to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Ghys that can be formulated in terms of semicomplete vector fields as follows: let X be a semicomplete holomorphic vector field on (Cn, 0) with isolated singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Is it true that J2X(0) ̸= 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' must the second jet of X at the singular point be different from zero ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Ghys’ original motivation seems to be related to problems about bounds for the dimension of automorphism group of compact complex manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' To be more precise, consider a compact complex manifold M and denote by Aut (M) the group of holomorphic diffeomorphisms of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' It is well known that Aut (M) is a finite dimensional complex Lie group whose Lie algebra can be identified with X (M), the space of all holomorphic vector fields defined on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' A too na¨ıve question, would be to wonder if the dimension of Aut (M) can be bounded by a function of the dimension of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' It turns out, however, that the dimension of the automorphism group of the Hirzebruch surface Fn is n + 5 provided that n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, already in the case of compact surfaces, the dimension of Aut (M) can be arbitrarily large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' However, analogous questions can be raised to better effect for specific classes of manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' For example, among projective manifolds with Picard group isomorphic to Z, Hwuang and Mok asked if there is a n-dimensional manifold whose dimension of the automorphism group exceeds the dimension of the automorphism group of CPn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' As a matter of fact, Ghys’ question is part of a general principle with vaguely stated as follows: semicomplete singularities cannot be “too degenerate”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Here it is convenient to explain how limiting the extent to which a semicomplete singularity can be degenerate becomes a useful tool GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS 39 to deal with the previous questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Consider a n-dimensional compact complex manifold M and let Aut (M) and X (M) be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Fix a point p ∈ M and let k ∈ N be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Finally, let Xk p(M) stand for the set of holomorphic vector fields with vanishing k-jet at p and denote by Jk p (M) the space of k-jets at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The natural mappings Xk p (M) → X (M) → Jk p (M) , give rise to a short exact sequence so that we have dim X (M) ≤ dim Xk p (M) + dim Jk p (M) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The dimensions of the jet spaces Jk p (M) are explicitly given in terms of k and of n = dim (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, if for some p ∈ M and k ∈ N, we can obtain bounds for dim Xk p (M) in terms of dim (M) then bounds for dim (Aut (M)) follow immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' For example, suppose that we happen to know that for a certain class of compact manifolds every singularity of a globally defined holomorphic vector field is necessarily isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Then, assuming Ghys conjecture holds, it follows that dim X3 p (M) = 0 and therefore the dimension of Aut (M) would be bounded by (n3 + 3n2 + 2n)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Of course, in general, non-isolated singularities also appear so that it is convenient to be able to handle them as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Aside from introducing the notion of semicomplete singularity, the content of [41] can fairly be summarized by the following theorem: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [41] Let X be a holomorphic semicomplete vector field on (C2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' If the origin is an isolated singular point for X, then J2 0X ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='10 relies on Camacho-Sad theorem on the existence of separatrices for foliations on (C2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Indeed, since the singular set of X is reduced to the origin, the restric- tion of X to any analytic invariant curve going through the origin cannot vanish identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Furthermore, this restriction is still a semicomplete vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Considering then the restric- tion of X to a separatrix, whose existence is ensured by Camacho-Sad theorem, the problem becomes essentially reduced to the one-dimensional situation (whether or not the separatrix is smooth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The resulting (one-dimensional) problem is settled in the same paper by direct methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The question on whether or not Ghys conjecture holds for semicomplete vector fields in higher dimensions is hence natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The first deep investigations involving semicomplete vector fields in higher dimensions were conducted by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Guillot in [16], and [17] (here “higher” means ≥ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The mentioned papers by Guillot contain, in particular, numerous examples of quadratic semi- complete vector fields exhibiting a wide range of geometric and dynamical behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Among these examples, we have already discussed the case of Halphen vector fields that have compli- cated dynamics and no (non-trivial) holomorphic/meromorphic first integral (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Moreover, Guillot’s work also make clear that in dimensions ≥ 3, an exhaustive classification of all semicomplete vector fields with zero linear part - paralleling the list provided in [13] - is unlikely to exist or, at least, it would be too long to be truly useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This is therefore a good moment to elaborate on the difficulties in extending to (C3, 0) the general classification results in dimension 2 of [41], [13], not to mention the more general results of [19] encompassing also meromorphic vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Indeed, whether or not obtaining these generalizations is a tall order, it certainly seems useful to explicitly list some of the new difficulties arising in dimensions greater than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Aside from the existence of core dynamics, that is a general difficulty already emphasized in this work, the following issues are worth mentioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 40 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REBELO AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REIS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The basic approach to Ghys conjecture stemming from [41] consists of finding a sep- aratrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Namely, the following holds: let X be a semicomplete (holomorphic) vector field on (Cn, 0) with an isolated singularity at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' If X possesses a separatrix, then J2 0 X ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' However, as previously seen, Gomez-Mont and Luengo [15] have proved that separatrices do not exist in general for germs of 1-dimensional foliations on (Cn, 0), n ≥ 3 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The examples provided in [15], however, are not semicomplete so that it is conceivable that all semicomplete vector field possesses a separatrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' While this seems to suggest that Ghys conjecture may be proved by showing that semicomplete vector fields do have separatrices, a direct approach to the latter question does not seem feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' A more promising point of view regarding item 2 above consists of noticing that the detailed classification of semicomplete vector fields in dimension 2, as developed in [13] or in [19], dispenses with Camacho-Sad theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In fact, these deeper analysis yield directly the classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Hence the existence of separatrices for semicomplete vector fields on (C2, 0) becomes a corollary, as opposed to a statement needed a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In dimension 2, a fundamental ingredient permeating virtually all works on singularities of vector fields or foliations is the resolution theorem of Seidenberg [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Since resolu- tions theorems for 1-dimensional foliations have been established in the past few years, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Section 4, this initial difficulty has now been overcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In fact, as far as semicom- plete vector fields are concerned, a totally faithful analogue of Seidenberg’s theorem is available in dimension 3 as will be seen below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Difficulties, however, are not limited to reduction of singularities procedures nor to the phenomenon of core dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' For example, assume our objective is to establish Ghys conjecture by means of proving the existence of separatrices (in which case the role played by core dynamics is significantly reduced, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Assume, in addition, that we are given a holomorphic foliation admitting a simple reduction of singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In dimension 3, the existence of saddle-node singularities appearing in the resolution procedure cannot easily be ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, codimension 2 saddle-nodes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' with two eigenvalues equal to zero) may appear and these singularities are still poorly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The remainder of this paper is to complement the above list with further comments and results, some of them proposing simpler approaches that can be effective pending on the specific application targeted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Concerning items 1 and 2, some partial results have been proved in [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In fact, recall that Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='8 states that in the case we are given two holomorphic vector fields X and Y yielding a representation of a Lie algebra of dimension 2 and not everywhere parallel, then they possess a common separatrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' By elaborating on this theorem, the following weaker version of Ghys conjecture in dimension 3 was proven in [47]: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [47] Consider a compact complex manifold M of dimension 3 and assume that the dimension of Aut (M) is at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let Z be an element of X (M) and suppose that p ∈ M is an isolated singularity of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Then J2(Z) (p) ̸= 0 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' the second jet of Z at the point p is different from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The reader will note that, as far as estimates on the dimension of automorphism groups are concerned, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='11 is as effective as an affirmative solution to Ghys conjecture in GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS 41 dimension 3 in the sense that if the additional assumption needed for Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='11 is not verified, then the dimension of Aut (M) is at most 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' One of the advantages of looking for bounds for the dimension of Aut (M) by means of local considerations is that the results obtained are essentially valid for open manifolds as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' For example, studying finite dimensional Lie group actions on Stein manifolds is an active topic in several complex variables whose roots lie in a classical work of Suzuki [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In this direction, our techniques yield: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [47] Let N denote a Stein manifold of dimension 3 and consider a finite dimensional Lie algebra G embedded in Xcomp(N) (the space of complete holomorphic vector fields on N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Assume that the dimension of G is at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' If Z is an element of G ⊆ X (M) possessing an isolated singular point p ∈ N, then the linear part of Z at p cannot vanish, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' p is a non-degenerate singularity of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' We may point out that the automorphism group of a Stein manifold is not a finite dimensional Lie group in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Indeed, even C2 has an infinite dimensional group of automorphisms with hardly any non-trivial structure of Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This difficulty is avoided in the statement of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='12 by the assumption that, from the beginning, we are dealing with some finite dimensional Lie algebra: owing to Lie theorem, such Lie algebra can always be integrated to yield a (complete) action of the corresponding Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This part of the statement actually holds for arbitrary complex manifolds of dimension 3 (whether or not they are compact or Stein) and parallels Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='11 in the sense that the second jet of a vector field Z ∈ G will never be zero at an isolated singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The role played by the Stein condition is to ensure that the first jet, rather than the second one, is different from zero at isolated singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' To close this paper, let us go back to resolution theorems as discussed in Section 4 and further sharpen the results under the additional assumption of having semicomplete vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' As already mentioned, resolution theorems always play a central role in singularity theory and sharp resolution statements exist for 1-dimensional foliations in dimension 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Yet, given their importance, it is convenient to have available the simplest possible resolution statements in every circumstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, it is natural to wonder if semicomplete singularities or other special classes of singular points allow for simpler resolution theorems facilitating a more detailed analysis of their structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' The possibility of having simpler resolution theorems valid for semicomplete singularities was also considered in [51] whose initial motivation was, in fact, to obtain a resolution theorem for semicomplete singularities that would faithfully parallel Seidenberg theorem for foliations on (C2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This type of statement is useful to approach problems such as Ghys conjecture or to investigate compact complex manifolds of dimension 3 equipped with holomorphic vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In this setting, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='13 below is proved in [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [51] Let X be a semicomplete vector field defined on a neighborhood of the origin of C3 and denote by F the holomorphic foliation associated with X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Then one of the following holds: (1) The linear part of X at the origin is nilpotent (non-zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (2) There exists a finite sequence of (standard) blow-ups along with transformed foliations F = F0 π1 ←− F1 π2 ←− · · · πr ←− Fr such that all of the singular points of Fr are elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Moreover, each blow-up map πi is centered in the singular set of the corresponding foliation Fi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In other words, the foliation F can be resolved by means of standard blow-ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 42 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REBELO AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REIS Let us emphasize that item 1 in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='13 involves the linear part of the vector field X rather than the linear part of the associated foliation F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In fact, it is the linear part of X that has to be (nilpotent) non-zero from the outset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Furthermore, this property is “universal” in the sense that it does not depend on any sequence of blow-ups/blow-downs carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, we can choose a “minimal model” for our manifold and the corresponding transform of X will still have non-zero nilpotent linear part at the corresponding point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' From what precedes, the following also deserves to be singled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [51] Let X be a semicomplete vector field defined on a neighborhood of (0, 0, 0) ∈ C3 and assume that the linear part of X at the origin is equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Then item (2) of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='13 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Accurate normal forms for persistent nilpotent singular points were provided in the same paper, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' However, not all of them need to be semicomplete or, indeed, realized as singularity of a complete flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Taking into account the global setting of complete vector fields, it is natural to wonder if there is, indeed, complete vector fields inducing a foliation with singular points that cannot be resolved by standard blow ups as in item (2) of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' As a matter of fact, these singularities do exist and an explicit example is provided by the polynomial vector field Z = x2 ∂ ∂x + xz ∂ ∂y + (y − xz) ∂ ∂z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Although Z is not complete on C3, it can be extended to a complete vector field defined on a suitable open manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In particular, the point corresponding to the origin of the above coordinates (x, y, z) constitutes a nilpotent singular point of Z that cannot be resolved by means of standard blow-ups with centers in the singular set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Finally, we might emphasize that the example above involves a complete vector field defined on an open manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' We might then ask if this phenomenon still occurs in the far more restric- tive context of compact manifolds of dimension 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Since in the compact case the completeness condition becomes automatic, we are simply asking whether or not there is a compact complex manifold of dimension 3 equipped with a (global) holomorphic vector field X which exhibits a singular point that cannot be resolved by means of standard blow ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' This time, the answer turns out to be negative as the following holds: Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Let F be the foliation associated with a vector field X globally defined on some compact complex manifold M of dimension 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Then every singular point of F can be resolved by a sequence of standard blow ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' In closing, let us just point out that both Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='14 and Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='15 are strictly speak- ing by-products of the methods used to prove Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='13 rather than formal consequences of the statement of this theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' References [1] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Barth, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Hulek, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Peters, & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Van de Ven, Compact complex surfaces, second edition, Springer- Verlag, Berlin, (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Briot & J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Bouquet, Propri´et´es des fonctions d´efinies par des ´equations diff´erentielles, Journal de l’Ecole Polytechnique, 36 (1856), 133-198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [3] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Camacho, Problems on Limit Sets of Foliations of Complex Projective Spaces, International Congress of Mathematicians (Kyoto), Springer Verlag, (1990), 1235-1239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [4] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Camacho, Quadratic forms and holomorphic foliations on singular surfaces, Matematische Annalen, 282, (1988), 177-184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [5] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Camacho & P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Sad, Invariant Varieties through Singularities of Holomorphic Vector Fields, Annals of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', 115 (1982), 579-595.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' GLOBAL DYNAMICS AND SINGULARITIES OF FOLIATIONS 43 [6] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Cano, Desingularization Strategies for Three Dimensional Vector Fields, Lecture Notes in Mathematics, 1259, Springer-Verlag, Berlin, (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [7] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Cano & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Cerveau, Desingularization of non-dicritical holomorphic foliations and existence of sepa- ratrices, Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', 169, (1992), 1-103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [8] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Cano, Reduction of the singularities of codimension one singular foliations in dimension three, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', 160, 3, (2004), 907-1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [9] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Cano, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Roche & M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Spivakovsky, Reduction of singularities of three-dimensional line foliations, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Serie A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Matematicas, February 2013, DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='1007/s13398-013-0117-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Dumitrescu, M´etriques riemanniennes holomorphes en petite dimension, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Fourier (Grenoble), 51, 6, (2001), 1663-1690.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [11] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Elizarov, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Il’yashenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Shcherbakov, & S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Voronin, Finitely generated groups of germs of one-dimensional conformal mappings and invariants for complex singular points of analytic foliations of the complex plane, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' in Soviet Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 14, (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [12] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Elshafei, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Ferreira & H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Reis, Geodesic completeness of pseudo and holomorphic Riemannian metrics on Lie groups, preprint available from arXiv: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='org/abs/2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='10873.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [13] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Ghys & J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Rebelo, Singularit´es des flots holomorphes II, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Fourier (Grenoble), 47, 4, (1997), 1117-1174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [14] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Griffiths & J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Harris, Principles of Algebraic Geometry, Wiley Classics Library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' John Wilet & Sons Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', New York (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [15] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Gomez-Mont & I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Luengo, Germs of holomorphic vector fields in C3 without a separatrix, Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', 109, 2 (1992), 211-219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [16] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Guillot, Semicompleteness of homogeneous quadratic vector fields, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Fourier (Grenoble), 56, 5, (2006), 1583-1615.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [17] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Guillot, Sur les ´equations d’Halphen et les actions de SL (2, C), Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' IHES, 105, 1, (2007), 221-294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [18] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Guillot, Meromorphic vector fields with single-valued solutions on complex surfaces, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', 354, (2019), 106742, 41 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Guillot & J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Rebelo, Semicomplete meromorphic vector fields on complex surfaces, Journal fur die reine und angewandte Mathematik, 667, (2012), 27-65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [20] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Halphen, Sur un syst`eme d’´equations diff´erentielles, Comptes Rendus Hebdomadaires de l’Acad´emie des Sciences, Vol XCII, 24, (1881), 1101-1102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [21] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Halphen, Sur certains syst`emes d’´equations diff´erentielles, Comptes Rendus Hebdomadaires de l’Acad´emie des Sciences, Vol XCII, 24, (1881), 1404-1406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Huddai-Verenov, A property of the solutions of a differential equation (Russian), Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Sbornik, 56 (98), 3, (1962), 301-308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [23] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Il’yashenko & S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Yakovenko, Lectures on analytic differential equations, Graduate Studies in Math- ematics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' 86, American Mathematical Society, Providence, RI, (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [24] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Il’yashenko, The topology of phase portraits of analytic differential equations in the complex projective plane (Russian), Trudy Sem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Petrovsk, 4, (1978), 83-136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (English), Sel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Sov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', 5, 2, (1986), 141-199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [25] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Ince, Ordinary Differential Equations, Dover Publications, New York, (1944).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [26] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Jouanolou, ´Equations de Pfaff alg´ebriques, Lect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Notes Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', 708, (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [27] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Jouanolou, Hypersurfaces solutions d’une ´equation de Pfaff analytique, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', 232, (1978), 239-245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [28] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Loray & J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Rebelo, Minimal, rigid foliations by curves in CPn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', 5, (2003), 147-201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [29] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Malgrange, Frobenius avec singularit´es, I: codimension un, Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' IHES, 46, (1976), 163-173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [30] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Marin & J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Mattei, Monodromy and topological classification of germs of holomorphic foliations, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' ´Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Sup´er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (4), 45, 3, (2012), 405-445.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [31] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Marin & J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Mattei, Topology of singular holomorphic foliations along a compact divisor, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Singul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', 9, (2014), 122-150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [32] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Marin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Mattei & E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Salem, Topological moduli space for germs of holomorphic foliations, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Notice IRMN, 23, (2020), 9228-9292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [33] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Marin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Mattei & E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Salem, Topological moduli space for germs of holomorphic foliations II: universal deformations, preprint, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='org/abs/2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='12688 [34] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Marin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Mattei & E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Salem, Topological moduli space for germs of holomorphic foliations III: complete families, preprint, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='org/abs/2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='07479 44 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REBELO AND H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' REIS [35] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Mattei & R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Moussu, Holonomie et int´egrales premi`eres, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' S´erie IV, 13, 4, (1980), 469-523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [36] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' McQuillan & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Panazzolo, Almost ´etale resolution of foliations, preprint IHES, IHES/M/09/51, (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [37] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' McQuillan & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Panazzolo, Almost ´etale resolution of foliations, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Differential Geometry, 95, (2013), 279-319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [38] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Nakai, Separatrizes for non solvable dynamics on (C, 0), Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Fourier, 44, (1994), 569-599.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [39] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Panazzolo, Resolution of singularities of real-analytic vector fields in dimension three, Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', 197, no 2 (2006), 167-289.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [40] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Piltant, An Axiomatic Version of Zariski’s Patching Theorem, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Cienc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Exactas Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' A Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' RACSAM, 107, 1, (2013), 91-121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [41] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Rebelo, Singularit´es des flots holomorphes, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Fourier (Grenoble), 46, 2, (1996), 411-428.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [42] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Rebelo, On transverse rigidity for singular foliations in (C2, 0), Ergod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' & Dynam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', 31, 3, (2011), 935-950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [43] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Rebelo & H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Reis, Local Theory of Holomorphic Foliations and Vector Fields, Lecture Notes available from arxiv (arXiv:1101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='4309).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [44] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Rebelo & H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Reis, Separatrices for C2-actions on 3-manifolds, Commentarii Mathematici Helvetici, 88, 3, (2013), 677-714.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [45] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Rebelo & H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Reis, Uniformizing complex ODEs and Applications, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Iberoam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', 30, 3, (2014), 799-874.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [46] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Rebelo & H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Reis, A note on integrability and finite orbits for subgroups of Diff (Cn, 0), Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Braz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (NS), 46, 3, (2015), 469-490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [47] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Rebelo & H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Reis, 2-dimensional Lie algebras and separatrices for vector fields on (C3, 0), Journal de Math´ematiques Pures et Appliqu´ees, 105, 2, (2016), 248-264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [48] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Rebelo & H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Reis, Discrete orbits, recurrence and solvable subgroups of Diff (C2, 0), Journal of Geometric Analysis, 27, 1, (2017), 1-55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [49] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Rebelo & H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Reis, On resolution of 1-dimensional foliations on 3-manifolds, Russian Mathematical Surveys, 76, 2, (2021), 291-355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [50] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Reis, Equivalence and semi-completude of foliations, Nonlinear Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Theory, Methods and Applica- tions, 64, 8, (2006), 1654-1665.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [51] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Reis, The geometry and dynamics of complex ordinary differential equations, Habilitation, University of Porto, Portugal (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [52] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Ribon, Recurrent orbits of subgroups of local complex analytic diffeomorphisms, Mathematische Zeitschrift, 285, (2017), 519-548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [53] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Shcherbakov, On the density of an orbit of a pseudogroup of conformal mappings and a generalization of the Hudai-Verenov theorem, Vestn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Mosk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=', 31, Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' I, (1982), 10-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [54] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Shcherbakov, Topological and analytic conjugation of non-commutative groups of conformal map- pings, Tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Semin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Petrovsk, 10, (1984), 170-192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [55] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Seidenberg, Reduction of singularities of the differential equation Ady=Bdx, American Journal of Mathematics, 90, (1968), 248-269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [56] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Suzuki, Sur les op´erations holomorphes de C et de C∗ sur un espace de Stein, S´eminaire F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Norguet, Springer LNM, 670, (1975-1976), 58-66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [57] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Teyssier, Germes de feuilletages pr´esentables du plan complexe, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Braz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' (NS), 46, 2, (2015), 275-329.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [58] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Touzet, Feuilletages holomorphes admettant une mesure transverse invariante, Annales de la Fac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' des Sciences de Toulouse, XXIV, 3, (2015), 523-541.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' [59] H, Zoladek, The monodromy group, Mathematics Institute of the Polish Academy of Sciences, Mathemat- ical Monographs (New Series), 67, Birkh¨auser Verlag, Basel, (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Institut de Math´ematiques de Toulouse ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' UMR 5219, Universit´e de Toulouse, 118 Route de Narbonne, F-31062 Toulouse, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Email address: rebelo@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='univ-toulouse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='fr Centro de Matem´atica da Universidade do Porto, Faculdade de Economia da Universidade do Porto, Portugal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content=' Email address: hreis@fep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} +page_content='pt' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE5T4oBgHgl3EQfTA_5/content/2301.05534v1.pdf'} diff --git a/BtE4T4oBgHgl3EQfeA0g/vector_store/index.faiss b/BtE4T4oBgHgl3EQfeA0g/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..38327bf01c4465b5aefe79127ef30d5a1da3de12 --- /dev/null +++ b/BtE4T4oBgHgl3EQfeA0g/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6dea8103fff2da5fbcf8442eaa2131432ca9ccea054994c5863966cfae8ad75f +size 4522029 diff --git a/CdFQT4oBgHgl3EQf_DcV/vector_store/index.faiss b/CdFQT4oBgHgl3EQf_DcV/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..f3fd68c924960fd5509e0ee3d464e13ce430fbcd --- /dev/null +++ b/CdFQT4oBgHgl3EQf_DcV/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5c7b38eb3d34066a64be845d286e9b48b2332c8bacac50fe3e9ee24bee28f379 +size 5373997 diff --git a/DNE0T4oBgHgl3EQfQQC5/content/2301.02191v1.pdf b/DNE0T4oBgHgl3EQfQQC5/content/2301.02191v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..dd87a755719832e801e939141736add1370c8c72 --- /dev/null +++ b/DNE0T4oBgHgl3EQfQQC5/content/2301.02191v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:269dda7389ef6e44079366b1d1da64ae6b4f4f36fbf6123ce9758f5000fa288f +size 3647817 diff --git a/DNE0T4oBgHgl3EQfQQC5/vector_store/index.faiss b/DNE0T4oBgHgl3EQfQQC5/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..e2b137868e1a48b6baa968de8deaaa5d892498f5 --- /dev/null +++ b/DNE0T4oBgHgl3EQfQQC5/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:df28ba6f2f731913e7bfda4521d7d3673e85cb4076fe158bbfd0b549d2894f39 +size 5898285 diff --git a/DNE0T4oBgHgl3EQfQQC5/vector_store/index.pkl b/DNE0T4oBgHgl3EQfQQC5/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..35a43c777eb4cb24d495a2427f87ffd01f2f059a --- /dev/null +++ b/DNE0T4oBgHgl3EQfQQC5/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c921c3ec8cc1a5d640ced80b0c8e636cb8785e8cfeb45472042d76812a7c2ecb +size 172883 diff --git a/G9E1T4oBgHgl3EQfXQT3/content/tmp_files/2301.03126v1.pdf.txt b/G9E1T4oBgHgl3EQfXQT3/content/tmp_files/2301.03126v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..419dc7d00021207286cc22ae9116311f259421ec --- /dev/null +++ b/G9E1T4oBgHgl3EQfXQT3/content/tmp_files/2301.03126v1.pdf.txt @@ -0,0 +1,5260 @@ +Statistical Inference for Ultrahigh Dimensional +Location Parameter Based on Spatial Median +Guanghui Chenga, +Liuhua Pengb, +Changliang Zouc +aGuangzhou Institute of International Finance, Guangzhou University, +bSchool of Mathematics and Statistics, The University of Melbourne, +c School of Statistics and Data Science, Nankai University +Abstract +Motivated by the widely used geometric median-of-means estimator in machine learning, +this paper studies statistical inference for ultrahigh dimensionality location parameter based +on the sample spatial median under a general multivariate model, including simultaneous +confidence intervals construction, global tests, and multiple testing with false discovery rate +control. To achieve these goals, we derive a novel Bahadur representation of the sample spa- +tial median with a maximum-norm bound on the remainder term, and establish Gaussian +approximation for the sample spatial median over the class of hyperrectangles. In addi- +tion, a multiplier bootstrap algorithm is proposed to approximate the distribution of the +sample spatial median. The approximations are valid when the dimension diverges at an +exponentially rate of the sample size, which facilitates the application of the spatial median +in the ultrahigh dimensional region. The proposed approaches are further illustrated by +simulations and analysis of a genomic dataset from a microarray study. +keywords: Bootstrap approximation; Gaussian approximation; high-dimensional; spatial me- +dian; FDR control +1 +Introduction +Geometric median-of-means (GMOM) has been widely used for robust estimation of multivariate +means, and it has been broadly adopted in machine learning (Minsker 2015, Hsu & Sabato 2016, +Prasad et al. 2020). The idea of GMOM is to first divide the data into disjoint subsamples +and calculate the empirical means of each of the subsamples. Then the GMOM estimator is +computed as the spatial median (also called geometric median) of the obtained empirical means. +The previous studies on the GMOM focused on establishing its non-asymptotic error bounds +1 +arXiv:2301.03126v1 [stat.ME] 9 Jan 2023 + +under certain heavy-tailed assumptions. Its distributional properties, which are essential for +statistical inference, remain unknown. +High-dimensional data with the dimension increases to infinity as the number of observa- +tions goes to infinity have been encountered in many scientific disciplines. There is a growing +evidence of the multivariate normal distribution is problematic to model high-dimensional data +due to the presents of heavy-tailedness and inadequate to accommodate tail dependence. For +example, the distributions of the microarray expression are observed to be non-normal and have +heavy tails even after log transformation in many gene expression data (Purdom & Holmes +2005, Wang, Peng and Li 2015). As another example, elliptical distributions, in particular the +multivariate t-distribution and symmetric multivariate normal inverse Gaussian distribution, +provided far superior models to the multivariate normal for daily and weekly US stock-return +data (McNeil et al. 2005). In such cases, the sample spatial median is favored against the sample +mean for estimating the location parameter. The above discussions strongly motivate studying +the spatial median under high-dimensionality, especially its distributional properties and the +implementation in statistical inference for high-dimensional location parameter. +Let X1, . . . , Xn be a sequence of independent and identically distributed (i.i.d.) p-dimensional +random vectors from a population X with cumulative distribution function FX in Rp. In this +paper, we work on a general multivariate model where X admits the following stochastic repre- +sentation: +X “ θ ` νΓU , +(1) +where θ is the location parameter, ν is a nonnegative univariate random variable and U is a +p-dimensional random vector with independent components. Model (1) covers many commonly +used multivariate models and distribution families, including the independent components model +(Yao et al. 2015) and the elliptical distribution family (Fang et al. 1990). We refer to Section 2 +for more detailed discussions. +Spatial median, an extension of the univariate median to multivariate distributions, was +proposed for robust inference of the location parameter (Haldane 1948, Weber 1929). The sample +spatial median ˆθn P Rp minimizes the empirical criteria function Lnpβq “ řn +i“1p}Xi´β}´}Xi}q, +2 + +where } ¨ } is the Euclidean norm. Equivalently, +ˆθn “ argmin +βPRp +Lnpβq “ argmin +βPRp +nÿ +i“1 +p}Xi ´ β} ´ }Xi}q . +(2) +The function Lnpβq is convex, and ˆθn is unique if the observations tXiun +i“1 are not concentrated +on a line in Rp when p ą 2 (Milasevic & Ducharme 1987). When the dimension p is fixed, the +spatial median has been well studied in the literature. We refer to Chapter 6.2 of Oja (2010) +for a nice review. +In the high-dimensional setting, where the dimension p diverges to infinity as the number +of observations n Ñ 8, there are several existing works that study the asymptotic properties +of the sample spatial median. +Zou et al. (2014) offered an expansion of ˆθn under elliptical +distributions with identical shape matrix, and Cheng et.al (2019) extended the result to a +general shape matrix. As a recent work, Li & Xu (2022) improved the expansion in Cheng et.al +(2019) with a smaller order remainder term under stronger conditions, and established a central +limit theorem for the squared Euclidean distance }ˆθn ´θ}2. In Zou et al. (2014) and Cheng et.al +(2019), they both require that p “ Opn2q. In addition, it is required in Li & Xu (2022) that p +diverges at the same rate as n. However, in modern areas such as genomics and proteomics, the +dimension of the data may grow exponentially with the sample size, which lies in the “ultrahigh +dimensional” region (Fan & Lv 2008). The previous works with restrictions on the polynomial +dimensionality limit the usage of the spatial median under ultrahigh-dimensionality. Moreover, +the previous results are all under elliptical distributions. Thus, it is of great importance to +establish asymptotic properties of the spatial median and investigate its applications under +ultrahigh dimensionality and beyond elliptical distributions. +In this paper, we first establish Gaussian and bootstrap approximations hit hyperrectangles +for the sample spatial median under the general model (1) beyond elliptical distributions, which +are valid when the dimension diverges exponential with the sample size. They serve as the +theoretical foundations of statistical inference for the location parameter based on the sample +spatial median under ultrahigh dimensionality. Consistent simultaneous confidence intervals +(SCIs) and global tests for the location parameters are established. We also study multiple +testing for every component of θ based on ˆθn. Motivated by simultaneous inference of θ, we +3 + +define a high-dimensional asymptotic relative efficiency of the sample spatial median relative +to the sample mean. Most importantly, our theoretical results guarantee the validity of the +proposed inferential methods for exponentially divergent p. The advantages of our proposed +approaches have been justified by simulations and a real data analysis. +The main contributions of this paper are summarized as follow. Firstly, we establish SCIs +for the location parameter θ based on the sample spatial median ˆθn, which is new in the +literature. The consistency of bootstrap approximation guarantees that the probability that +the SCIs cover all components of the location parameter approaches the nominal confidence +level under ultrahigh dimensionality. We also propose a novel test for ultrahigh dimensional +location parameter based on the maximum-norm of the sample spatial median. The proposed +test not only maintains nominal significance level asymptotically for exponentially divergent p, +but also is more powerful under sparse alternatives compared to those based on L2-norms (Li +& Xu 2022, Wang, Peng and Li 2015). As another major inference, we study multiple testing +for every component of the location parameter, and the false discovery rate (FDR) can be +well controlled combined with the Benjamini-Hochberg procedure based on the sample spatial +median, which extends the existing methods based on the sample mean (Liu and Shao 2014). +In all inferential methods, the procedures based on the sample spatial median advances those +based on the sample mean for heavy-tailed distributions. +Secondly, this paper serves as the first work that provides Gaussian and bootstrap approxi- +mations for the sample spatial median under ultrahigh dimensionality. Gaussian and bootstrap +approximations for high-dimensional sample mean have received extensive attraction in the last +decade. Chernozhukov et al. (2013) and Chernozhukov, Chetverikov and Kato (2017) established +Gaussian and bootstrap approximations for the maxima of a sum of centered independent ran- +dom vectors under Kolmogorov distance and on hyperrectangles, respectively. See also Chen +(2018), Chernozhukov et al. (2019) and Chernozhukov et al. (2020) for related works. Compared +to the sample mean, which has a simple linear form, the theoretical difficulty for the sample +spatial median lies in that it does not enjoy an explicit form. This issue is addressed by deriving +a novel Bahadur representation of the sample spatial median with a maximum-norm bound on +the remainder term, which extends the results of Zou et al. (2014), Cheng et.al (2019) and Li & +Xu (2022) under elliptical distributions and polynomial dimensionality. Moreover, our results +4 + +can be applied to the GMOM under reasonable conditions, and thus enhance the practice usage +of GMOM. +Thirdly, we propose a novel multiplier bootstrap method for the sample spatial median. In- +stead of multiplying on the loss function, which is generally the case for M-estimator (Imaizumi +& Otsu 2021), the multiplier is applied on the centralized Xi. Specifically, the bootstrap version +of ˆθn is defined as ˜θn “ argminβPRd řn +i“1 }ZipXi ´ ˆθnq ´ β}, where Z1, . . . , Zn are the multipli- +ers. The multiplier bootstrap is consistent under ultrahigh dimensionality thanks to this novel +formulation. This is, however, different from the multiplier bootstrap method for the sample +mean, which again has an explicit form (Chernozhukov et al. 2013, Chernozhukov, Chetverikov +and Kato 2017). +The rest of the paper is organized as follows. Section 2 introduces model and assumptions. +Section 3 establishes Gaussian and bootstrap approximations to the distribution of the sample +spatial median. Statistical inference for the location parameter based on the sample spatial +median is presented in Section 4. Section 5 reports numerical results including simulations and +a real data analysis. Preliminary lemmas and proofs of main results are presented in Appendix +A of the supplementary material. Proofs of preliminary lemmas and additional simulations are +given in Appendices B and C of the supplementary material. +Notation: Denote |x|8 “ maxp|x1|, . . . , |xd|q as the maximum-norm of x “ px1, . . . , xdqJ. +Denote an À bn if an ď Cbn for a positive constant C, and an — bn means an À bn and bn À an. +For α ą 0, let ψαpxq “ exppxαq ´ 1 be a function defined on r0, 8q. Then the Orlicz norm +}¨}ψα of a random variable X is defined as }X}ψα “ inf tt ą 0, Etψα p|X|{tqu ď 1u . We use trp¨q +to denote the trace operator for square matrices. Moreover, we denote Ip as the p ˆ p identity +matrix. For a, b P R, we write a ^ b “ minpa, bq. +2 +Model and assumptions +In this paper, we consider a general multivariate model for the distribution FX such that Xi +admits the following stochastic representation: +Xi “ θ ` νiΓUi , +(3) +5 + +where θ is the location parameter, Γ is a nonrandom and invertible p ˆ p matrix, Ui is a p- +dimensional random vector with independent standardized components, and νi is a nonnegative +univariate random variable independent with the spatial sign of Ui. The distribution of Xi +depends on Γ through the shape matrix Ω “ ΓΓJ. +Remark 1. Model (3) covers many commonly used multivariate models and distribution fam- +ilies. +First, the independent components model (Yao et al. 2015) follows (3) with νi being +a nonnegative constant. Second, model (3) also includes elliptical distributions by choosing +Ui „ Np0, Ipq and νi “ ξi{}Ui} for some nonnegative random variable ξi independent of Ui. In +this case, νi is independent of the spatial sign of Ui, but not Ui. The independent components +model has received great extension in high-dimensional data analysis as well as signal process- +ing and machine learning (Hyv¨arinen et al. 2001). In addition, the elliptical distribution family +covers many non-Gaussian distributions such as multivariate t-distribution, multivariate logistic +distribution, and so on. It is commonly adopted in the literature on studying the sample spatial +median (Cheng et.al 2019, Li & Xu 2022, Zou et al. 2014). In terms of the GMOM, if the data +are from the independent components model, the subsample means satisfy model (3) clearly. In +addition, some subfamilies of elliptical distributions are closed under convolution, and thus the +subsample means also follow model (3). Our results can be applied to the GMOM estimator +directly in those cases. +For i “ 1, . . . , n, and k “ 1, 2, 3, 4, denote +Wi “ SpXi ´ θq and Ri “ }Xi ´ θ} +(4) +as the spatial-sign and radius of Xi ´ θ, where SpXq “ }X}´1XIpX ‰ 0q is the multivariate +sign function with Ip¨q being the indicator function. Thus, ˆθn satisfies řn +i“1 SpXi ´ ˆθnq “ 0 . +Denote Ui “ pUi,1, . . . , Ui,pqJ, we impose the following three conditions. +Condition C.1. Ui,1, . . . , Ui,p are i.i.d. symmetric random variables with EpUi,jq “ 0, EpU 2 +i,jq “ +1, and }Ui,j}ψα ď c0 with some constant c0 ą 0 and 1 ď α ď 2. +Condition C.2. The moments ζk “ EpR´k +i +q for k “ 1, 2, 3, 4 exist for large enough p. In +addition, there exist two positive constants b and ¯B such that b ď lim supp EpRi{?pq´k ď ¯B for +k “ 1, 2, 3, 4. +6 + +Condition C.3. The shape matrix Ω “ pωjℓqpˆp satisfies trpΩq “ p and it belongs to the +following class: +Upa0ppq, m, ¯ +Mq “ +# +Ω : m ď ωjj ď ¯ +M, +pÿ +ℓ“1 +|ωjℓ| ď a0ppq, +for all j “ 1, . . . , p ++ +, +where m ď ¯ +M are bounded positive constants. +Remark 2. In Condition C.1, the symmetric assumption is to ensure that θ in model (3) co- +incides with the population spatial median, which minimizes Lpβq “ Ep}X ´ β} ´ }X}q. It is +obvious that Condition C.1 is satisfied by elliptical distributions with Ui „ Np0, Ipq. The condi- +tion }Ui,j}ψα ď c0 implies that Ui,j has a sub-exponential distribution. It is worth highlighting +that with slight modification of the proofs of main theorems, the i.i.d. condition on Ui,1, . . . , Ui,p +can be weaken by replacing Condition C.1 with the following assumption: Ui,1, . . . , Ui,p are in- +dependent symmetric random variables with EpUi,jq “ 0, EpU 2 +i,jq “ 1 for all j “ 1, . . . , p, and +sup1ďjďp }Ui,j}ψα ď c0 with some constant c0 ą 0 and 1 ď α ď 2. +Remark 3. The condition b ď lim supp EpRi{?pq´k ď ¯B indicates that ζk — p´k{2 for k “ +1, 2, 3, 4. +It is introduced to avoid Xi from concentrating too much near θ. +For elliptical +distributions, it is a generalization of Assumption 1 of Zou et al. (2014), which is satisfied +by many common distributions. +For the independent components model, Condition C.2 is +equivalent to that b ď lim supp Ep}ΓUi}{?pq´k ď ¯B . According to Lemma A2 in Appendix +A, Ep}ΓUi}kq “ pk{2t1 ` op1qu for k “ 1, 2, 3, 4. Then the Cauchy-Schwarz inequality implies +that Ep}ΓUi}´kq ě tEp}ΓUi}kqu´1 “ p´k{2t1 ` op1qu , from which we know Ep}ΓUi}´kq Á p´k{2. +Furthermore, denote Γj as the jth row of Γ, then by the inequality of harmonic and quadratic +means, +p2}ΓUi}´4 “ +" +p +pΓ1Uiq2 ` ¨ ¨ ¨ ` pΓpUiq2 +* +ď pΓ1Uiq´4 ` ¨ ¨ ¨ ` pΓpUiq´4 +p +. +It follows that Ep}ΓUi}´4q À p´2 if EtpΓ1Uiq´4u, . . . , EtpΓpUiq´4u are uniformly bounded, and +from which Ep}ΓUi}´kq À p´k{2 by Jensen’s inequality. Thus, Condition C.2 is satisfied by the +independent components models as long as Γ1Ui, . . . , ΓpUi are not concentrating too much near +0. See also discussions in Cardot et al. (2013) on similar conditions. +7 + +Remark 4. It is noticed that the shape matrix Ω is only well defined up to a scalar multiple, +the condition trpΩq “ p is used to regularize Ω to make model (3) identifiable. +The class +Upa0ppq, m, ¯ +Mq covers a wide range of symmetric square matrices, and it is commonly adopted +in the literature on high-dimensional analysis. For example, a similar matrix class is introduced +in Bickel & Levina (2008). The condition m ď ωjj ď ¯ +M requires bounded diagonal elements. +The order of a0ppq, which will be specified later, controls the orders of the off-diagonal elements +of Ω. +3 +Gaussian and bootstrap approximations +3.1 +Bahadur representation and Gaussian approximation +In this section, we establish Gaussian approximation for ˆθn, which is valid when p diverges +exponentially over n. The following lemma offers a Bahadur representation of ˆθn, and it severs +as the foundation of the Gaussian approximation result in Theorem 1. +Lemma 1. (Bahadur representation) Assume Conditions C.1, C.2 and C.3 with a0ppq — p1´δ +for some positive constant δ ď 1{2 hold. If log p “ opn1{3q and log n “ opp1{3^δq, then +n1{2pˆθn ´ θq “ n´1{2ζ´1 +1 +nÿ +i“1 +Wi ` Cn , +where |Cn|8 “ Optn´1{4 log1{2pnpq ` p´p1{6^δ{2q log1{2pnpqu. +Remark 5. To the best of our knowledge, Lemma 1 serves as the first result that offers the +Bahadur representation of the sample spatial median with a maximum-norm bound on the +remainder term. In Zou et al. (2014) and Cheng et.al (2019), the same expansion with the +remainder term Cn satisfies }Cn} “ oppζ´1 +1 q was obtained, and their result was improved to +}Cn} “ opp1q in Li & Xu (2022), by replacing ζ1 with n´1 řn +i“1 R´1 +i +in the linear term, but +under a more restricted condition that p and n are of the same order. It is worth noticing that +the previous results (Cheng et.al 2019, Li & Xu 2022, Zou et al. 2014) are all derived under +elliptical distributions. +Let Are “ tśp +j“1raj, bjs : ´8 ď aj ď bj ď 8, j “ 1, . . . , pu be the class of rectangles in +8 + +Rp. With the Bahadur representation in Lemma 1 on hand, we establish the following Gaussian +approximation result for ˆθn over hyperrectangles. +Theorem 1. (Gaussian approximation) Assume Conditions C.1, C.2 and C.3 with a0ppq — p1´δ +for some positive constant δ ď 1{2 hold. If log p “ opn1{5q and log n “ opp1{3^δq, then +ρnpAreq “ sup +APAre +ˇˇˇPtn1{2pˆθn ´ θq P Au ´ P pG P Aq +ˇˇˇ Ñ 0 +as n Ñ 8, where G „ Np0, ζ´2 +1 Bq with B “ EpW1W J +1 q. +The Gaussian approximation for ˆθn indicates that the probabilities Ptn1{2pˆθn ´ θq P Au can +be approximated by that of a centered Gaussian random vector with covariance matrix ζ´2 +1 B +for hyperrectangles A P Are. Theorem 1 allows for an exponentially divergent p, which fits the +ultrahigh dimensional setting. Compared to the asymptotic normality of }ˆθn ´ θ}2 in Li & Xu +(2022), in which p is assumed to have the same order as n, the Gaussian approximation result +in Theorem 1 requires much weaker conditions on the rates of n and p. +Remark 6. Let Bjℓ be the pj, ℓqth element of B. According to Lemma A4 (iii) in Appendix +A, ζ´2 +1 Bjℓ “ ζ´2 +1 p´1ωj,ℓ ` Opp´δ{2q for all 1 ď j, ℓ ď p. Thus, the covariance matrix of G in +Theorem 1 is asymptotically proportional to the shape matrix Ω. +Remark 7. As the sample spatial median is a special M-estimator, Gaussian approximation for +M-estimator in Imaizumi & Otsu (2021) is potentially applicable to the spatial median under +high-dimensionality. However, it is worth highlighting that the results in Imaizumi & Otsu +(2021) cannot be applied to our framework. To be precise, Assumption 1 (A3) in Imaizumi & +Otsu (2021) assumes that there exist constants C ą 0 and α P p0, 2q such that log Npε, Θ, }¨}q ď +Cε´α holds for all ε P p0, 1q, where Θ is the parameter space, and Npε, Θ, } ¨ }q is the ε-covering +number of Θ under the Euclidean norm } ¨ } (van der Vaart & Wellner 1996). When Θ is a +compact subset of Rp, Npε, Θ, } ¨ }q is of order Opε´pq. In this case, log Npε, Θ, } ¨ }q ď Cε´α +cannot be satisfied when p Ñ 8. Thus, our theoretical findings are independent of those in +Imaizumi & Otsu (2021). +Theorem 1 immediately implies the following corollary since the Kolmogorov distance of +sup-norm is a subset of Are corresponding to max-hyperrectangles in Rp. +9 + +Corollary 1. Under the conditions assumed in Theorem 1, as n Ñ 8, +ρn “ sup +tPR +ˇˇˇPpn1{2|ˆθn ´ θ|8 ď tq ´ Pp|G|8 ď tq +ˇˇˇ Ñ 0. +3.2 +Multiplier bootstrap approximation +Theorem 1 allows us to approximate the distribution of n1{2pˆθn ´θq by that of G hit hyperrect- +angles, where G „ Np0, ζ´2 +1 Bq. However, it cannot be used directly in statistical inference for θ +as the quantity ζ1 and the matrix B depend on the underlying distribution FX and are thus un- +known. To solve this issue, we propose an easy-to-implement bootstrap method to approximate +the distribution of n1{2pˆθn ´ θq. +Let Z1, . . . , Zn be a sequence of i.i.d. random variables with mean zero and unit variance. +Define the bootstrap version of the sample spatial median as +˜θn “ argmin +βPRd +nÿ +i“1 +}ZipXi ´ ˆθnq ´ β} . +(5) +Then, the distribution of n1{2˜θn conditional on X1, . . . , Xn is used to approximate that of +n1{2pˆθn´θq. This algorithm is called the multiplier bootstrap, and Z1, . . . , Zn are the multiplier +weights. +Regarding the proof of Lemma A5 in Appendix B, it is preferred that the multiplier weights +Z1, . . . , Zn are bounded and satisfy EpZ´2 +i +q ă 8. Thus, we choose the Rademacher variables as +the multipliers (Chernozhukov et al. 2019), that is, PpZi “ 1q “ PpZi “ ´1q “ 1{2. +Theorem 2. (Bootstrap approximation) Under the conditions assumed in Theorem 1, +ρMB +n +pAreq “ sup +APAre +ˇˇˇPtn1{2pˆθn ´ θq P Au ´ P˚pn1{2˜θn P Aq +ˇˇˇ Ñ 0 +in probability as n Ñ 8, where P˚ denotes the conditional probability given X1, . . . , Xn. +Under the same conditions on the divergence rates of n and p as in Theorem 1, Theorem +2 validates that conditional on X1, . . . , Xn, the distribution of the bootstrap sample spatial +median ˜θn approximates that of ˆθn consistently over hyperrectangles. +Remark 8. The proof of Theorem 2 is nontrivial and does not follow directly from existing +10 + +results since ˜θn has no explicit form, which is different from the multiplier bootstrap methods +for high-dimensional sample mean that have been analysed in the literature. +The key step +in the proof is to obtain a Bahadur representation of ˜θn similar as ˆθn in Lemma 1. Specif- +ically, we show that n1{2˜θn “ n´1{2ζ´1 +1 +řn +i“1 ZiWi ` ˜Cn with | ˜Cn|8 “ Optn´1{4 log1{2pnpq ` +p´p1{6^δ{2q log1{2pnpqu in Lemma A5 in Appendix A. +The next corollary is an immediate consequence of Theorem 2. +Corollary 2. Under the conditions assumed in Theorem 2, as n Ñ 8, +ρMB +n +“ sup +tPR +ˇˇˇPtn1{2|ˆθn ´ θ|8 ď tu ´ P˚pn1{2|˜θn|8 ď tq +ˇˇˇ Ñ 0 in probability. +4 +Statistical inference +The Gaussian and multiplier bootstrap approximations for the sample spatial median enable +many statistical inferential methods for ultrahigh dimensional population location parameter. +In this section, we present the following statistical inferences: simultaneous confidence intervals +(SCIs) and global tests for the population location parameter, multiple testing for every com- +ponent of θ, and high-dimensional asymptotic relatively efficient of the sample spatial median +compared to the sample mean. +4.1 +Simultaneous confidence intervals +We are interested in building SCIs for all components of θ “ pθ1, . . . , θpqJ. Corollary 2 motivates +the following way of constructing SCIs for θ. Given a nominal confidence level 1 ´ τ, define the +set Cτ as +Cτ “ +! +θ P Rp, n1{2|ˆθn ´ θ|8 ă qB +1´τ +) +, +where qB +1´τ is the p1´τqth quantile of n1{2|˜θn|8 given X1, . . . , Xn. Denote ˆθn “ pˆθn,1, . . . , ˆθn,pqJ, +the confidence intervals are rθ´ +n,j, θ` +n,js for j “ 1, . . . , p, where +θ´ +n,j “ ˆθn,j ´ n´1{2qB +1´τ and θ` +n,j “ ˆθn,j ` n´1{2qB +1´τ. +11 + +The next theorem shows that Cτ preserves the nominal simultaneous confidence level 1 ´ τ +asymptotically under ultrahigh dimensionality. +Theorem 3. Suppose the conditions of Theorem 2 hold, then Ppθ P Cτq Ñ 1 ´ τ as n Ñ 8. +Equivalently, Ppθj P rθ´ +n,j, θ` +n,js for all 1 ď j ď pq Ñ 1 ´ τ as n Ñ 8. +Remark 9. Unlike the fixed dimensional setting, n1{2|˜θn|8 and n1{2|ˆθn ´ θ|8 are maxima of +divergent numbers of variables, and their quantiles are generally divergent as p Ñ 8. Thus, +Theorem 3 is not a direct consequence of Corollary 2. +To ascertain the consistency of Cτ +theoretically, we show that, with probability approaching one, qB +1´τ is bounded by two quantiles +of n1{2|ˆθn ´θ|8 with quantile levels close enough to 1´τ using an anti-concentration inequality +for divergent random sequences. +Remark 10. The Gaussian approximation for the sample mean ¯Xn “ n´1 řn +i“1 Xi (Cher- +nozhukov et al. 2013, Chernozhukov, Chetverikov and Kato 2017, Chernozhukov et al. 2019) +indicate that if log p “ opn1{5q, +sup +tPR +ˇˇˇPpn1{2| ¯Xn ´ θ|8 ď tq ´ Pp|G0|8 ď tq +ˇˇˇ Ñ 0 +(6) +as n Ñ 8 under some moderate conditions, where G0 „ Np0, Σq with Σ “ EpXXJq. Define +X˚ +i “ ZipXi ´ ¯Xnq for i “ 1, . . . , n, where Z1, . . . , Zn are the Rademacher weights. Denote +¯X˚ +n “ n´1 řn +i“1 X˚ +i , it has been shown in Chernozhukov et al. (2019) that +suptPR +ˇˇPpn1{2| ¯Xn ´ θ|8 ď tq ´ P˚pn1{2| ¯X˚ +n|8 ď tq +ˇˇ Ñ 0 +(7) +in probability as n Ñ 8 when log p “ opn1{5q. Based on (7), define +C1 +τ “ +! +θ P Rp, n1{2| ¯Xn ´ θ|8 ă qB1 +1´τ +) +, +where qB1 +1´τ is the p1 ´ τqth quantile of n1{2| ¯X˚ +n|8 conditional on X1, . . . , Xn. Then C1 +τ is also +an asymptotic 1 ´ τ SCIs for θ. +Based on the discussion in Section 4.4, Cτ has advantage +(relative shorter intervals) over C1 +τ under heavy-tailed distributions. We refer to Section 5.1 for +finite-sample justifications on this. +12 + +4.2 +Global tests for high-dimensional location parameters +In this section, we propose a novel approach for global tests on high-dimensional location pa- +rameters. Let θ0 be a known p-dimensional vector, we are interested in testing +H0 : θ “ θ0 versus H1 : θ ‰ θ0. +(8) +Theorems 1 and 2 motivate us proposing a maximum-norm type test statistic. Define +Tn “ n1{2|ˆθn ´ θ0|8 +(9) +as the test statistic, and H0 is rejected when Tn is larger than a critical value. We can use +the multiplier bootstrap to approximate the distribution of Tn under H0. Specifically, with a +nominal significance level τ, the null hypothesis is rejected if Tn ą qB +1´τ. Theorem 3 guarantees +that the test based on Tn maintains nominal significance level asymptotically under ultrahigh +dimensionality, that is, PpTn ą qB +1´τ | H0q Ñ τ as n Ñ 8 when log p “ opn1{5q. +Remark 11. An alternative test for (8) can be constructed based on ¯Xn by defining the test +statistic as TMean “ n1{2| ¯Xn ´ θ0|8. Then, the null hypothesis is rejected if TMean ą qB1 +1´τ. The +test based on Tn can be deemed as a nonparametric extension of the test based on TMean . As +ˆθn is more efficient than ¯Xn for simultaneous inference of θ under heavy-tailed distributions +as discussed in Section 4.4, we expect that the proposed test based on Tn is more powerful +than that based on TMean in those cases. This has been reflected by the simulation results in +Appendix C of the supplementary material. +The next theorem summarises the asymptotic power of the proposed test based on Tn. +Theorem 4. Suppose the conditions of Theorem 2 hold. For any given 0 ă τ ă 1, if |θ´θ0|8 ě +C log1{2pτ ´1qn´1{2 log1{2pnpq for some large enough constant C ą 0, then PpTn ą qB +1´τ | H1q Ñ +1 as n Ñ 8. +Theorem 4 indicates that the test based on Tn achieves consistency when the maximum +element of n1{2|θ ´ θ0| has a magnitude much large than log1{2pτ ´1q log1{2pnpq for a fixed +significant level τ. +13 + +Remark 12. Wang, Peng and Li (2015) proposed a L2-norm type test (WPL test) for (8) with +θ0 “ 0 based on TWPL “ řn +i“1 +ři´1 +j“1 W J +i Wi. It has been argued in Wang, Peng and Li (2015) +and Li & Xu (2022) that the signal of the WPL test is determined by the magnitude of }θ}, +which is the L2-norm of θ. As a contrast, the power of the test based on Tn depends on |θ|8. +Thus, the proposed test based on Tn is expected to be more powerful under sparse alternatives, +when θ contains only a limited number of non-zero components and its maximum element has +certain order of magnitude. In such cases, }θ} is not big enough for the rejection of the WPL +test. See Appendix C in the supplementary material and Section 5.3 for numerical justifications. +4.3 +Multiple testing with FDR control in large-scale tests +Multiple testing with false discovery rate (FDR) control has been applied to many real problems, +such as detecting differentially expressed genes in genomic study. +In this section, we study +multiple testing for every component of θ based on the spatial median with the Benjamini and +Hochberg (B-H) method for FDR control. For j “ 1, . . . , p, we are interested in testing +H0j : θj “ θ0,j versus H1j : θj ‰ θ0,j +simultaneously, where θ0,1, . . . , θ0,p are given values. +Define the test statistics as +Tn,j “ n1{2pˆθn,j ´ θ0,jq{sn,j +for j “ 1, . . . , p, where s2 +n,j “ ˆζ´2 +1 +ˆBjj with ˆζ1 “ n´1 řn +i“1 }Xi ´ ˆθn}´1, and ˆBjj is the jth +diagonal element of ˆB “ n´1 řn +i“1 }Xi ´ ˆθn}´2pXi ´ ˆθnqpXi ´ ˆθnqJ. +According to the proof of Theorem 5 in Appendix A, Tn,j converges in distribution to a +standard normal under H0j for j “ 1, . . . , p. Thus, we utilise the standard normal distribution to +estimate the marginal p-values. For j “ 1, . . . , p, define the p-value for H0j as Pj “ 2´2Φp|Tn,j|q. +Denote Pp1q ď ¨ ¨ ¨ ď Pppq be the ordered p-values, and define +ˆk “ max +␣ +j “ 0, . . . , p : Ppjq ď τj{p +( +14 + +for a pre-specific significance level τ. Then, the B-H procedure rejects the null hypotheses for +which Pj ď Ppˆkq. Denote HR “ tj : Pj ď Ppˆkqu as the set of indices j such that H0j is rejected +by the B-H method, and let |HR| be the cardinality of HR that equals the total number of +rejected null hypotheses. +Let H0 Ă t1, . . . , pu be the set of indices j corresponding to the true null hypotheses H0j. +The false discovery proportion (FDP) and false discovery rate (FDR) of the B-H method are +defined as +FDPM “ |H0 X HR| +|HR| _ 1 +and FDRM “ EpFDPMq. +Regarding that Tn,1, . . . , Tn,p are dependent, we impose the following condition on the weak +dependence between any two components of Wi. Define prjℓqpˆp “ tdiagpBqu´1{2BtdiagpBqu´1{2 +as the correlation matrix, where diagpBq is the diagonal matrix of B. +Condition C.4. Suppose max1ďj,ℓďp |rjℓ| ď r with some constant 0 ă r ă 1. In addition, +řp +j“1 Iprjℓ “ 0q “ Oppηq for some constant 0 ă η ă p1 ´ rq{p1 ` rq. +Similar conditions are assumed in Liu and Shao (2014) and Belloni et.al (2018). Let p0 “ |H0| +be the number of true null hypotheses and Bjj be the jth diagonal element of B. +Theorem 5. Suppose Condition C.4 and the conditions of Theorem 1 hold. In addition, there +exists H Ă t1, . . . , pu such that H “ +␣ +j : ζ1B´1{2 +jj +n1{2|θj ´ θ0,j| ě 2 log1{2ppq +( +and |H| ě +log log p Ñ 8 as p Ñ 8. Assume that the number of false null hypotheses p1 ď pϖ for some +0 ă ϖ ă 1. Then, FDRM{pτp0{pq Ñ 1 as n Ñ 8. +Theorem 5 shows the B-H procedure based on P1, . . . , Pp controls the FDR asymptotically, +and it extends Theorem 4.1 in Liu and Shao (2014) to spatial median-based test statistic. +4.4 +High-dimensional asymptotic relative efficiency +As two candidate estimators of the location parameter θ, it is of interest to study the asymptotic +relative efficiency (ARE) of the sample spatial median ˆθn relative to the sample mean ¯Xn. +When p is fixed, for spherical multivariate normal distribution, Brown (1983) showed that the +asymptotic efficiency of ˆθn relative ¯Xn, denoted as AREpˆθn, ¯Xnq, exceeds the usual univariate +15 + +case 2{π. In addition, AREpˆθn, ¯Xnq increases as the dimension increases, and it approaches to 1 +as p tends to be sufficient large (Magyar & Tyler 2011). However, when p Ñ 8, the ARE is not +straightforward to quantify as there are no obvious “final” limit distributions for ˆθn and ¯Xn. +Motivated by the discussions in Sections 4.1 and 4.2, we compare ˆθn and ¯Xn in terms of their +efficiencies in simultaneous inference for θ, which are determined by the variations of |ˆθ ´ θ|8 +and | ¯Xn ´ θ|8. According to Corollary 1 and (6), we define the high-dimensional ARE of ˆθn +compared to ¯Xn in simultaneous inference for θ as +AREpˆθn, ¯Xnq “ Varp|G0|8q{Varp|G|8q , +(10) +which approximates Varp| ¯Xn ´ θ|8q{Varp|ˆθn ´ θ|8q. If limpÑ8 AREpˆθn, ¯Xnq ą 1, we say that +ˆθn is more efficient than ¯Xn in simultaneous inference for θ under high-dimensionality. +As discussed in Remark 6, G „ Np0, ζ´2 +1 Bq with ζ´2 +1 Bjℓ “ ζ´2 +1 p´1ωiℓ for all 1 ď j, ℓ ď p. +Moreover, we can show that Σjℓ “ Epν2 +i qωjℓ ` Opp´1{2q similar to the proof of Lemma A3 in +Appendix B of the supplementary material, where Σjℓ is the pj, ℓqth element of Σ. Thus, both +the covariance matrix Σ and ζ´2 +1 B are proportional to Ω asymptotically, and AREpˆθn, ¯Xnq is +approximately Epν2 +i qζ2 +1p. +As Σ and ζ´2 +1 B are rarely known in practice, we use bootstrap approximation to estimate +the value of Varp|G0|8q{Varp|G|8q. Combining Corollary 2 and (7), we propose using +Var˚p| ¯X˚ +n|8q{Var˚p|˜θn|8q, +to estimate AREpˆθn, ¯Xnq. +Example 1. Suppose X1, . . . Xn are i.i.d. from Npθ, Ipq, then ν2 +i follows a chi-square distribution +with p degrees of freedom. It follows that Epν2 +i q “ p and Epν´1 +i +q “ Γpp{2 ´ 1{2q{t21{2Γpp{2qu, +where Γp¨q is the gamma function.. So the ARE is AREpˆθn, ¯Xnq “ ptΓpp{2´1{2qu2{t21{2Γpp{2qu2. +Using Stirling’s formula, limpÑ8 AREpˆθn, ¯Xnq “ 1. Thus, for high-dimensional Gaussian data, +the sample spatial median has the same asymptotically efficiency as the sample mean in simul- +taneous inference for θ. +Example 2. When the data are from the multivariate t-distribution with degrees of freedom +v ą 2 and shape matrix Ω “ Ip, ν2 +i {p „ Fp,v, where Fp,v is the F distribution with parameters +16 + +p and v. Then, Epν2 +i q “ pv{pv ´ 2q and Epν´1 +i +q “ Γpv{2 ` 1{2qΓpp{2 ´ 1{2q{tv1{2Γpv{2qΓpp{2qu. +Thus, the ARE is AREpˆθn, ¯Xnq “ pv ´ 2q´1ptΓpv{2 ` 1{2qΓpp{2 ´ 1{2qu2{tΓpv{2qΓpp{2qu2. +It is clear that AREpˆθn, +¯Xq ą 1 for large enough p. +In addition, limpÑ8 AREpˆθn, ¯Xnq “ +2pv ´ 2q´1tΓpv{2 ` 1{2qu2{tΓpv{2qu2 ą 1 . Thus, for high-dimensional t-distribution, the sample +spatial median is asymptotically more efficient than the sample mean in simultaneous inference +for θ. +Figure 1 plots the simulated values of Varp| ¯Xn|8q{Varp|ˆθn|8q with a range of dimensions +and sample sizes under different models. For Gaussian data, the relative efficiency kept increas- +ing in p, and it approached 1 as p getting larger. For the data simulated from multivariate +t-distribution, the relative efficiency was greater than 1 for all combinations of n and p. This in- +dicates that the sample spatial median is more efficiency than the sample mean for t-distribution. +The results were consistent under different covariance structure considered in the simulation. +0.94 +0.96 +0.98 +1.00 +0 +100 +200 +300 +400 +p +Efficiencies +Gaussian, ρ=0 +1.5 +1.6 +1.7 +1.8 +1.9 +2.0 +2.1 +0 +100 +200 +300 +400 +p +Efficiencies +t5, ρ=0 +0.90 +0.95 +1.00 +0 +100 +200 +300 +400 +p +Efficiencies +Gaussian, ρ=0.8 +1.4 +1.5 +1.6 +1.7 +1.8 +1.9 +0 +100 +200 +300 +400 +p +Efficiencies +t5, ρ=0.8 +n +20 +40 +80 +Figure 1: Finite sample relative efficiency of |ˆθn|8 compared to | ¯Xn|8 based on 5000 replications, +the data are generated from multivariate normal distribution (Gaussian) and t-distribution with +5 degrees of freedom (t5). The shape matrix Ω “ pρ|j´ℓ|qpˆp with ρ “ 0 and 0.8. +17 + +5 +Numerical studies +In this section, we report Monte Carlo simulations on simultaneous confidence intervals and +multiple testing with FDR control, along with a real data analysis, to demonstrate the per- +formance of the proposed approaches. Additional simulations on global tests can be found in +Appendix C of the supplementary material. In the simulations, all results were based on 2500 +replications. In the bootstrap implementation, the number of bootstrap iterations was set to +B “ 400. +5.1 +Simulations on simultaneous confidence intervals +We first examine the performance of the SCIs based on ˆθn, and compare it with the SCIs based +on ¯Xn. The sample size n is taken to be 100 or 200, and the dimensions p “ 100 and 1000 +are considered for each sample size. Two types of commonly used elliptical distributions are +considered: (I) the multivariate normal distribution Npθ, Σq; (II) the multivariate t-distribution +with 3 degrees of freedom, mean vector θ, and covariance matrix Σ. In addition, we include the +following independent components model: (III) Xi “ θ ` Σ1{2Zi, where each component of Zi +are i.i.d. from the standard Laplace distribution. We set Σ “ pρ|j´ℓ|q with ρ “ 0, 0.2, 0.5 and +0.8. To save space, we present the results for ρ “ 0 and 0.8 here. The results for ρ P t0.2, 0.5u +are similar and are reported in the supplementary material. We consider both sparse and dense +case scenarios for θ: (i) θ1 “ p2, ´2, 3, 0, . . . , 0q; (ii) θ2 “ p0.2, . . . , 0.2tp{4u, 0, . . . , 0q. Here t¨u is +the floor function. +Table 1 reports the coverage probability and median length of the SCIs based on ˆθn, the +results of the SCIs based on ¯Xn are presented in parentheses. For Models I and II from elliptical +distributions, we observe that the SCIs based on ˆθn and ¯Xn both achieve satisfying coverage +probability for different choices for ρ, θ, n and p. For the data simulated from the multivariate +normal distribution, the median length of the SCIs based on ˆθn is very close to that of the +the SCIs based on ¯Xn. These results indicate that ˆθn has similar asymptotic efficiency as ¯Xn +in simultaneous inference for θ under high-dimensional Gaussian model as discussed in Section +4.4. For the multivariate t-distribution, the SCIs based on ˆθn is much narrower than the SCIs +based on ¯Xn. These results suggest that the SCIs based on ˆθn is more efficient than the SCIs +18 + +based on ¯Xn for multivariate t-distribution, which is heavy-tailed. This is consistent with the +asymptotic analysis in Section 4.4. Moreover, the results for Model III, which does not belong to +the elliptical distribution family, shows the robustness of the SCIs based on the spatial median, +and it performs similar to the SCIs based on the sample mean. We also note that the median +length of the SCIs decreases when n increases or p decreases for each model. +Table 1: Coverage probability (in %) and median length of the SCIs based on ˆθn, the results of +the SCIs based on ¯Xn are in parentheses. +θ “ θ1 +θ “ θ2 +Coverage probability +Median length +Coverage probability +Median length +Model +ρ +n +p +90% +95% +90% +95% +90% +95% +90% +95% +I +0 +100 +100 89.6 (89.9) 94.4 (94.4) 0.65 (0.65) 0.69 (0.69) +88.9 (88.8) 94.1 (93.9) 0.65 (0.65) 0.69 (0.69) +1000 89.5 (89.6) 94.7 (94.4) 0.77 (0.77) 0.80 (0.80) +89.5 (89.5) 94.0 (94.0) 0.77 (0.77) 0.81 (0.80) +200 +100 89.8 (89.8) 95.1 (95.1) 0.46 (0.46) 0.49 (0.49) +88.6 (88.8) 94.4 (94.7) 0.46 (0.46) 0.49 (0.49) +1000 89.7 (89.7) 94.4 (94.6) 0.55 (0.55) 0.57 (0.57) +89.1 (89.2) 94.7 (94.6) 0.55 (0.55) 0.57 (0.57) +0.8 100 +100 89.1 (88.7) 94.6 (94.6) 0.64 (0.63) 0.68 (0.67) +88.4 (88.6) 93.7 (94.1) 0.64 (0.63) 0.68 (0.67) +1000 88.4 (88.4) 93.8 (93.7) 0.76 (0.76) 0.80 (0.79) +89.0 (89.2) 94.6 (94.6) 0.76 (0.76) 0.80 (0.79) +200 +100 90.5 (90.1) 95.2 (94.9) 0.45 (0.45) 0.48 (0.48) +89.6 (89.6) 94.0 (94.1) 0.45 (0.45) 0.48 (0.48) +1000 90.4 (90.4) 94.5 (94.4) 0.54 (0.54) 0.56 (0.56) +88.4 (88.5) 93.6 (93.8) 0.54 (0.54) 0.56 (0.56) +II +0 +100 +100 89.7 (88.6) 94.7 (93.7) 0.71 (1.05) 0.75 (1.11) +88.8 (88.8) 94.5 (94.2) 0.71 (1.05) 0.75 (1.11) +1000 89.4 (91.0) 95.8 (95.0) 0.84 (1.25) 0.88 (1.30) +89.1 (89.0) 94.4 (94.5) 0.84 (1.25) 0.88 (1.31) +200 +100 88.6 (89.1) 94.2 (95.1) 0.50 (0.76) 0.53 (0.81) +89.5 (89.7) 94.4 (94.8) 0.50 (0.76) 0.53 (0.80) +1000 89.6 (88.7) 94.8 (94.6) 0.59 (0.90) 0.62 (0.94) +90.1 (89.5) 94.8 (93.9) 0.59 (0.90) 0.62 (0.94) +0.8 100 +100 89.1 (90.7) 94.4 (94.9) 0.69 (1.02) 0.74 (1.09) +89.4 (89.7) 94.2 (94.4) 0.69 (1.02) 0.74 (1.09) +1000 89.3 (89.1) 94.6 (94.4) 0.83 (1.23) 0.87 (1.29) +89.8 (88.8) 94.7 (94.4) 0.83 (1.23) 0.87 (1.29) +200 +100 87.6 (87.7) 93.4 (93.6) 0.49 (0.73) 0.52 (0.78) +90.3 (90.1) 94.9 (95.2) 0.49 (0.73) 0.52 (0.78) +1000 88.7 (89.7) 94.7 (94.6) 0.59 (0.88) 0.61 (0.92) +90.2 (90.8) 94.7 (95.7) 0.59 (0.89) 0.61 (0.93) +III +0 +100 +100 +89.8 (89.4) 94.6 (94.5) 0.65 (0.66) 0.69 (0.70) +89.1 (89.0) 94.4 (94.4) 0.65 (0.66) 0.69 (0.70) +1000 88.3 (88.2) 93.6 (93.7) 0.78 (0.78) 0.82 (0.82) +89.1 (89.0) 94.2 (93.8) 0.78 (0.78) 0.82 (0.82) +200 +100 90.6 (91.1) 95.0 (95.0) 0.46 (0.46) 0.49 (0.49) +90.6 (90.1) 95.2 (95.2) 0.46 (0.46) 0.49 (0.49) +1000 90.1 (90.4) 95.0 (94.6) 0.55 (0.55) 0.57 (0.58) +88.7 (89) +93.6 (93.8) 0.55 (0.55) 0.57 (0.58) +0.8 100 +100 +90.4 (89.7) 95.0 (94.8) 0.63 (0.63) 0.68 (0.68) +89.0 (88.9) 95.0 (94.9) 0.63 (0.63) 0.67 (0.68) +1000 88.7 (88.9) 93.8 (94.0) 0.77 (0.77) 0.80 (0.80) +89.0 (89.0) 94.6 (94.3) 0.76 (0.76) 0.80 (0.80) +200 +100 88.8 (89.1) 94.2 (94.0) 0.45 (0.45) 0.48 (0.48) +90.2 (89.7) 94.8 (95.0) 0.45 (0.45) 0.48 (0.48) +1000 90.0 (90.3) 95.0 (95.0) 0.54 (0.54) 0.57 (0.57) +88.8 (89.1) 94.2 (94.1) 0.54 (0.54) 0.57 (0.57) +5.2 +Simulations on multiple testing with FDR control +In this section, we examine the performance of the sample spatial median-based B-H method +introduced in Section 4.3, and compare it to the B-H procedure based on the sample mean with +p-values calculated from Np0, 1q in Liu and Shao (2014). We set θ0,j “ 0 for all j “ 1, . . . , p. +The data are generated from Models I and II with p “ 1000. +For θ “ pθ1, . . . , θpqJ, let +θj “ 2plog p{nq1{2 for 1 ď j ď p1 and θj “ 0 for pp1 ` 1q ď j ď p, where p1 “ 0.1p. +Table 2 reports the empirical FDR and power for the sample spatial median-based (FDRM +and powerM) and the sample mean-based (FDRA and powerA) B-H procedures (Liu and Shao +19 + +2014) with nominal level α “ 0.1 and 0.2. The results indicate that the FDR are well controlled +by both methods. For the multivariate normal distribution, the B-H procedures based on the +spatial median and the sample mean have similar performance. However, the sample spatial +median-based B-H method outperforms the sample mean-based B-H procedure in terms of +empirical power under multivariate t-distribution, which is heavy-tailed. +Table 2: Empirical FDR and power for the spatial median-based (FDRM and powerM) and the +sample mean-based (FDRA and powerA) in Liu and Shao (2014) via B-H procedures. +α “ 0.1 +α “ 0.2 +Model +ρ +n +FDRM FDRA powerM powerA +FDRM FDRA powerM powerA +I +0 +50 +0.124 +0.124 +0.996 +0.996 +0.224 +0.222 +0.999 +0.999 +100 +0.107 +0.106 +0.997 +0.997 +0.202 +0.201 +0.999 +0.999 +0.2 50 +0.125 +0.124 +0.996 +0.996 +0.224 +0.223 +0.999 +0.999 +100 +0.107 +0.106 +0.997 +0.997 +0.202 +0.201 +0.999 +0.999 +0.5 50 +0.125 +0.124 +0.996 +0.996 +0.225 +0.223 +0.999 +0.999 +100 +0.107 +0.105 +0.997 +0.997 +0.202 +0.201 +0.999 +0.999 +0.8 50 +0.127 +0.124 +0.996 +0.996 +0.227 +0.223 +0.999 +0.999 +100 +0.108 +0.105 +0.997 +0.997 +0.204 +0.199 +0.999 +0.999 +II +0 +50 +0.117 +0.099 +0.984 +0.728 +0.215 +0.193 +0.992 +0.805 +100 +0.103 +0.088 +0.987 +0.710 +0.197 +0.179 +0.994 +0.795 +0.2 50 +0.117 +0.098 +0.984 +0.727 +0.215 +0.194 +0.992 +0.805 +100 +0.103 +0.087 +0.987 +0.709 +0.198 +0.179 +0.994 +0.795 +0.5 50 +0.118 +0.099 +0.984 +0.727 +0.216 +0.194 +0.992 +0.803 +100 +0.103 +0.087 +0.987 +0.708 +0.198 +0.178 +0.994 +0.794 +0.8 50 +0.120 +0.098 +0.984 +0.724 +0.218 +0.192 +0.992 +0.800 +100 +0.104 +0.087 +0.987 +0.705 +0.199 +0.177 +0.994 +0.791 +5.3 +Real data analysis +Type 2 diabetesis a disease in which the body becomes resistant to normal effects of insulin +and gradually loses the capacity to produce enough insulin. +Because skeletal muscle is the +main tissue for insulin-stimulated glucose disposal, skeletal muscle insulin resistance is com- +monly viewed as the critical component of whole-body insulin resistance, and thus is crit- +ical to the pathogenesis of Type 2 diabetes. +To investigate the effects of insulin on gene +expression in skeletal muscle, a microarray study was performed in 15 diabetic patients us- +ing the Affymetrix Hu95A chip of muscle biopsies both before and after insulin treatment +(Wu et al. 2007). +In this paper, we are interested in the gene expression alteration, that +20 + +is, the change of the gene expression level, due to the treatment. The data are available at +https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE22309. The data were normal- +ized by quantile normalization by the normalizeQuantiles function in the limma R package. +Follow Wang, Peng and Li (2015), we focused on 2547 curated gene sets with at least 15 genes, +which are from the C2 collection of the GSEA online pathway databases. The gene expression +values are consolidated by taking the average when multiple probes are associated with the same +gene. +We implemented the Median test based on Tn on the 2519 gene sets. This is equivalent to +testing whether the median change vector of gene expression levels is equal to 0. The number +of bootstrap iterations is B “ 105. With the Bonferroni correction, there are 1242 gene sets +identified as significant at 5% level. For comparison, we applied the WPL test (Wang, Peng +and Li 2015) and the CQ test (Chen and Qin 2010) on the same gene sets. For the WPL test, +1060 gene sets are selected as significant; and for the CQ test, 630 gene sets are identified as +significant. Out of the 630 gene sets selected by the CQ test, 605 of them are also identified by +our proposed method, and 629 of them are identified by the WPL test. It has been argued in +Wang, Peng and Li (2015) that some gene expression levels have heavy tails as their kurtosises +are much larger than the kurtosis of a normal distribution, 3. Thus, the methods based on the +spatial median (Median test and the WPL test) are expected to be more robust and efficient +than those based on moments (CQ test). In addition, out of the 1060 gene sets identified by +the WPL test, 958 of them are significant based on our proposed approach. +As argued in Remark 12, the Median test based on Tn is more powerful in detecting strong +sparse signal compared to the WPL test. To see this, we look into the following three gene sets: +(1) ZHAN MULTIPLE MYELOMA UP; +(2) MIKKELSEN MEF HCP WITH H3K27ME3; +(3) JAZAG TGFB1 SIGNALING VIA SMAD4 UP. +The p-values of the WPL test for these three gene sets are 0.41, 0.31, 0.27, respectively. However, +the p-values of the Median test are all less than 1.0 ˆ 10´5 with B “ 105 bootstrap iterations +for these three gene sets. Figure 2 plots the SCIs for the spatial median vectors of the change +of gene expression levels for these three gene sets. The confidence intervals that do not cover +0 are colored in red. It is very clear that the only one or two big values in the spatial median +21 + +results in a rejection of the Median test, while the signals from other dimensions are not strong +enough to land a rejection by the the WPL test. +Finally, we use the spatial median-based B-H procedure to perform multiple testing with +FDR control on the three gene sets to detect differentially expressed genes (DEG), which is one +of the most important targets in genomic analysis. Table 3 reports the detected differentially +expressed genes (DEG) in each gene set with nominal level α “ 0.1, along with the corresponding +marginal p-value Pj “ 2 ´ 2Φp|Tn,j|q and the confidence interval in the SCIs for the selected +genes. It can be seen that for all the selected genes, the marginal p-values are very small, and +the corresponding confidence intervals do not cover 0. +Table 3: Detected differentially expressed genes (DEG) by the spatial median-based B-H proce- +dure for three gene sets with α “ 0.1; “p-value” refers to the marginal p-value Pj “ 2´2Φp|Tn,j|q, +and “CI” refers to the confidence interval in the SCIs for the selected genes. +Gene set +DEG +p-value +CI +ZHAN MULTIPLE MYELOMA UP +CDKN1A +0.00082 +(0.234, 0.550) +MIKKELSEN MEF HCP WITH H3K27ME3 +MYOD1 +ă 0.00001 (0.433, 0.791) +JAZAG TGFB1 SIGNALING VIA SMAD4 UP +HDAC4 +0.00058 +(0.254, 0.644) +6 +Discussion +In this paper, we established one-sample and two-sample Gaussian and bootstrap approxima- +tions for ultrahigh dimensional sample spatial median under a general model beyond elliptical +distributions. It is of interest to study whether our results are potentially extendable to some +other distribution families. +We leave this to a future work. +In addition, the proposed test +based on the maxima of the sample spatial median is more powerful under sparse alternatives +compared to those based on L2-norms. It is well known that the L2-norm type tests are more +powerful under dense alternatives. Thus, it is of interest to consider combining the test based +on the maximum-norm and L2-norm, which could be potentially powerful under both sparse +and dense alternatives. We also leave this to a future study. +22 + +ZHAN_MULTIPLE_MYELOMA_UP +0 +20 +40 +60 +−0.2 +0.0 +0.2 +0.4 +Gene Index +Spatial Median +MIKKELSEN_MEF_HCP_WITH_H3K27ME3 +0 +100 +200 +300 +−0.3 +0.0 +0.3 +0.6 +Gene Index +Spatial Median +JAZAG_TGFB1_SIGNALING_VIA_SMAD4_UP +0 +20 +40 +60 +80 +−0.25 +0.00 +0.25 +0.50 +Gene Index +Spatial Median +Figure 2: Simultaneous Confidence intervals (SCIs) for spatial medians of three gene sets. +23 + +Supplementary material +The supplementary material includes all the technical proofs and some additional numerical +results. +References +Belloni, A., Chernozhukov, V., Chetverikov, D., Hansen, C. & Kato, K. (2018), ‘High- +dimensional econometrics and generalized gmm’, arXiv p. 1806.01888. +Bickel, P. J. & Levina, E. (2008), ‘Covariance regularization by thresholding’, Ann. Statist. +36, 2577–2604. +Brown, B. (1983), ‘Statistical uses of the spatial median’, J. R. Statist. Soc. B 45, 25–30. +Cardot, H., C´enac, P. & Zitt, P.-A. (2013), ‘Efficient and fast estimation of the geometric median +in hilbert spaces with an averaged stochastic gradient algorithm’, Bernoulli 19, 18–43. +Chen, S. X. & Qin, Y. (2010), ‘A two-sample test for high-dimensional data with applications +to gene-set testing’, Ann. Statist. 38, 808–835. +Chen, X. (2018), ‘Gaussian and bootstrap approximations for high-dimensional U-statistics and +their applications’, Ann. Statist. 46, 642–678. +Cheng, G., Liu, B., Peng, L., Zhang, B. & Zheng, S. (2019), ‘Testing the equality of two high- +dimensional spatial sign covariance matrices’, Scand. J. Statist. 46, 257–271. +Chernozhukov, V., Chetverikov, D. & Kato, K. (2013), ‘Gaussian approximations and multiplier +bootstrap for maxima of sums of high-dimensional random vectors’, Ann. Statist. 41, 2786– +2819. +Chernozhukov, V., Chetverikov, D. & Kato, K. (2017), ‘Central limit theorems and bootstrap +in high dimensions’, The Annals of Probability 45, 2309–2352. +Chernozhukov, V., Chetverikov, D. & Kato, K. (2019), ‘Improved central limit theorem and +bootstrap approximation in high dimensions’, arXiv p. 1912.10529. +24 + +Chernozhukov, V., Chetverikov, D., Kato, K. & Koike, Y. (2020), ‘Nearly optimal central limit +theorem and bootstrap approximations in high dimensions’, arXiv p. 2012.09513. +Fan, J. & Lv, J. (2008), ‘Sure independence screening for ultrahigh dimensional feature space’, +J. R. Statist. Soc. B 70, 849–911. +Fang, K. W., Kotz, S. & Ng, K. W. (1990), Symmetric multivariate and related distributions, +Boca Raton, FL: CRC Press. +Haldane, J. B. S. (1948), ‘Note on the median of a multivariate distribution’, Biometrika 35, 414– +417. +Hsu, D. & Sabato, S. (2016), ‘Loss minimization and parameter estimation with heavy tails’, J. +Mach. Learn. Res. 17, 1–40. +Hyv¨arinen, P., Karhunen, J. & Oja, E. (2001), Independent Component analysis, New York: +Wiley. +Imaizumi, M. & Otsu, T. (2021), ‘On gaussian approximation for m-estimator’, arXiv +p. 2012.15678v2. +Koike, Y. (2021), ‘Notes on the dimension dependence in high-dimensional central limit theorems +for hyperrectangles’, Japanese Journal of Statistics and Data Science 1, 257–297. +Li, W. & Xu, Y. (2022), ‘Asymptotic properties of high-dimensional spatial median in elliptical +distributions with application’, Journal of Multivariate Analysis 190, 104975. +Liu, W. & Shao, Q.-M. (2014), ‘Phase transition and regularized bootstrap in large scale t-tests +with false discovery rate control’, Ann. Statist. 42, 2003–2025. +Magyar, A. & Tyler, D. E. (2011), ‘The asymptotic efficiency of the spatial median for elliptically +symmetric distributions’, Sankhya B 73, 165–192. +McNeil, A. J., Frey, R. & Embrechts, P. (2005), Quantitative Risk Management: Concepts, +Techniques and Tools, Princeton, NJ: Princeton University Press. +Milasevic, P. & Ducharme, G. R. (1987), ‘Uniqueness of the spatial median’, Ann. Statist. +15, 1332–1333. +25 + +Minsker, S. (2015), ‘Geometric median and robust estimation in banach spaces’, Bernoulli +21, 2308–2335. +Oja, H. (2010), Multivariate nonparametric methods with R: An approach based on spatial signs +and ranks, Lecture Notes in Statistics, Springer, New York. +Prasad, A., Suggala, A. S., Balakrishnan, S. & Ravikumar, P. (2020), ‘Robust estimation via +robust gradient estimation’, J. R. Statist. Soc. B 82, 601–627. +Purdom, E. & Holmes, S. P. (2005), ‘Error distribution for gene expression data’, Statistical +Applications in Genetics and Molecular Biology 4, 1–35. +van der Vaart, A. W. & Wellner, J. A. (1996), Weak Convergence and Empirical Processes: +With Applications to Statistics, Springer. +Wang, L., Peng, B. & Li, R. (2015), ‘A high-dimensional nonparametric multivariate test for +mean vector’, J. Am. Statist. Assoc. 110, 1658–1669. +Weber, A. (1929), Uber Den Standort der Industrien (Alfred Weber?s Theory of the Location of +Industries), Chicago, IL: Univ. Chicago Press. +Wu, X., Wang, J., Cui, X., Maianu, L., Rhees, B., Rosinski, J., So, W. V., Willi, S. M., Osier, +M. V., Hill, H. S., Page, G. P., Allison, D. B., Maritin, M. & Garvey, W. T. (2007), ‘The +effect of insulin on expression of genes and biochemical pathways in human skeletal muscle’, +Endocrine 31, 5–17. +Yao, J., Zheng, S. & Bai, Z. (2015), Sample covariance matrices and high-dimensional data +analysis, Cambridge University Press, Cambridge. +Zou, C., Peng, L., Feng, L. & Wang, Z. (2014), ‘Multivariate sign-based high-dimensional tests +for sphericity’, Biometrika 101, 229–236. +26 + +Supplement to “Statistical Inference for Ultrahigh +Dimensional Location Parameter Based on Spatial +Median” +Appendix A: Technical Proofs +We first introduce and recall some notation. For a d1 ˆ d2 matrix M “ pmjℓqd1ˆd2, its matrix +ϱ-norm is }M}ϱ “ supt}Mx}ϱ : }x}ϱ “ 1u. Specifically, the 1-, 2-, and 8-norms of M are +}M}1 “ max1ďℓďd2 +řd1 +j“1 |mjℓ|, }M}2 “ tλmaxpMJMqu1{2, and }M}8 “ max1ďjďd1 +řd2 +ℓ“1 |mjℓ|. +The Frobenius norm of M is }M}F “ třd1 +j“1 +řd2 +ℓ“1 m2 +jℓu1{2. +Define a random p ˆ p matrix Q “ n´1 řn +i“1 R´1 +i WiW J +i +such that EpQq “ EpR´1 +i WiW J +i q, +and denote Qjℓ as the pj, ℓqth element of Q. Denote E˚p¨q and Var˚p¨q be the expectation and +variance conditional on X1, . . . , Xn, respectively. Recall that Wi,j is the jth element of Wi for +i “ 1, . . . , n and j “ 1, . . . , p; ωjℓ is the pj, ℓqth element of Ω; and Γj is the jth row of Γ. Finally, +we will denote various positive absolute constants by C1, C2, C3, . . . without mentioning this +explicitly. +A.1 +Preliminary lemmas +In this section. we present several preliminary lemmas, whose proof can be found in online +Appendix B. +Lemma A1. (Concentration of norms) Suppose that Conditions C.1 and C.3 hold with a0ppq — +p1´δ for some positive constant δ ď 1{2. Then, for sufficient large p, there exist positive con- +stants c1 and c2 such that +P +! +p ´ ϵpp1`δq{2 ď }U1}2 ď p ` ϵpp1`δq{2) +ě +1 ´ c1 expt´c2pδα{p4α`4qu +and +P +␣ +p1 ´ ϵqtrpΩq ď }ΓU1}2 ď p1 ` ϵqtrpΩq +( +ě +1 ´ c1 expt´c2pδα{p4α`4qu +27 + +for any fixed 0 ă ϵ ă 1. +Lemma A2. Suppose that Conditions C.1, C.2 and C.3 hold with a0ppq — p1´δ for some positive +constant δ ď 1{2. Then, for any i “ 1, . . . , n, +(i) Ep}Ui}4q “ pEpU 4 +i,jq ` ppp ´ 1q, +Ep}Ui}6q +“ +pEpU 6 +i,jq ` 3ppp ´ 1qEpU 4 +i,jq ` ppp ´ 1qpp ´ 2q, +Ep}Ui}8q +“ +pEpU 8 +i,jq ` 4ppp ´ 1qEpU 6 +i,j1q ` 3ppp ´ 1qtEpU 4 +i,j1qu2 +`3ppp ´ 1qEpU 4 +i,jq ` ppp ´ 1qpp ´ 2qpp ´ 3q . +In addition, Ep}Ui}2kq “ pk ` Oppk´1q and Ep}U}kq “ pk{2 ` Oppk{2´1q for any positive integer +k. +(ii) Ep}ΓUi}4q “ p2 ` Opp2´δq, Ep}ΓUi}6q “ p3 ` Opp3´δq. In addition, Ep}ΓUi}q “ p1{2 ` +Opp1{2´δq and Ep}ΓUi}3q “ p3{2 ` Opp3{2´δq. +(iii) Et}ΓSpUiq}2u “ 1 ` Opp´1{2q and Et}ΓSpUiq}4u “ 1 ` Opp´1{3q. +(iv) Epν´k +i +q À ζkpk{2 for k “ 1, 2, 3. +Lemma A3. Suppose Conditions C.1, C.2 and C.3 with a0ppq — p1´δ for some positive constant +δ ď 1{2 hold. Define a random pˆp matrix Q “ n´1 řn +i“1 R´1 +i WiW J +i +and let Qjℓ be the pj, ℓqth +element of Q. Then, +(i) |Qjℓ| À ζ1p´1|ωjℓ| ` Oppζ1n´1{2p´1 ` ζ1p´7{6 ` ζ1p´1´δ{2q. +(ii) Qjℓ “ Q0,jℓ ` Oppζ1p´7{6 ` ζ1p´1´δ{2q, where Q0,jℓ is the pj, ℓqth element of +Q0 “ n´1p´1{2 +nÿ +i“1 +ν´1 +i +tΓSpUiqutΓSpUiquJ. +In addition, Q0 satisfies +trrEpQ2 +0q ´ tEpQ0qu2s “ Opn´1p´1q. +Lemma A4. Suppose Conditions C.1, C.2 and C.3 with a0ppq — p1´δ for some positive constant +δ ď 1{2 hold. Then, +(i) Etpζ´1 +1 Wi,jq4u À ¯ +M2 and Etpζ´1 +1 Wi,jq2u Á m for all i “ 1, . . . , n and j “ 1, . . . , p. +(ii) }ζ´1 +1 Wi,j}ψα À ¯B for all i “ 1, . . . , n and j “ 1, . . . , p. +28 + +(iii) EpW 2 +i,jq “ p´1ωjj ` Opp´1´δ{2q for j “ 1, . . . , p and EpW 2 +i,jq “ p´1ωjℓ ` Opp´1´δ{2q for +1 ď j ‰ ℓ ď p. +(iv) if log p “ opn1{3q, +ˇˇˇˇˇn´1{2 +nÿ +i“1 +ζ´1 +1 Wi +ˇˇˇˇˇ +8 +“ Optlog1{2pnpqu and +ˇˇˇˇˇn´1 +nÿ +i“1 +pζ´1 +1 Wiq2 +ˇˇˇˇˇ +8 +“ Opp1q . +Lemma A5. Suppose the conditions of Theorem 2 hold, then +n1{2˜θn “ n´1{2ζ´1 +1 +nÿ +i“1 +ZiWi ` ˜Cn , +(S.11) +where | ˜Cn|8 “ Optn´1{4 log1{2pnpq ` p´p1{6^δ{2q log1{2pnpqu. +The following lemma is Nazarov’s inequality, and its proof can be found in Chernozhukov, +Chetverikov and Kato (2017). +Lemma A6 (Nazarov’s inequality). Let Y0 “ pY0,1, . . . , Y0,pqJ be a centered Gaussian random +vector in Rp and EpY 2 +0,jq ě b for all j “ 1, . . . , p and some constant b ą 0, then for every y P Rp +and a ą 0, +PpY0 ď y ` aq ´ PpY0 ď yq À a log1{2ppq . +A.2 +Proof of main results +Proof of Lemma 1. As θ is a location parameter, we assume θ “ 0 without loss of generality. +Then, Wi “ SpXiq “ }Xi}´1Xi “ }ΓUi}´1ΓUi for i “ 1, . . . , n. The sample spatial median ˆθn +satisfies +nÿ +i“1 +SpXi ´ ˆθnq “ +nÿ +i“1 +Xi ´ ˆθn +}Xi ´ ˆθn} +“ +nÿ +i“1 +Wi ´ R´1 +i +ˆθn +}Wi ´ R´1 +i +ˆθn} +“ 0 , +which is is equivalent to +n´1 +nÿ +i“1 +pWi ´ R´1 +i +ˆθnqp1 ´ 2R´1 +i W J +i ˆθn ` R´2 +i }ˆθn}2q´1{2 “ 0 +as W J +i Wi “ 1. +Under Condition C.2, ζk “ EpR´k +i +q “ Opp´k{2q for k “ 1, 2, 3, 4. +In addition, Lemma +29 + +A3 indicates that Qjℓ “ Q0,jℓ ` Oppζ1p´7{6 ` ζ1p´1´δ{2q, where Q0,jℓ is the pj, ℓqth element +of Q0 “ n´1p´1{2 řn +i“1 νitΓSpUiqutΓSpUiquJ . In addition, Q0 satisfies trrEpQ2 +0q ´ tEpQ0qu2s “ +Opn´1p´1q. Thus, from the similar procedure as in the proof of Lemma 1.2 of Cheng et.al (2019), +we can show that +}ˆθn} “ Oppζ´1 +1 n´1{2q. +Then, for i “ 1, . . . , n, we have |R´1 +i W T +i ˆθn| ď R´1 +i }ˆθn} “ Oppn´1{2q and R´2 +i ||ˆθn||2 “ Oppn´1q. +By the first-order Taylor expansion, the above equation can be rewritten as +n´1 +nÿ +i“1 +pWi ´ R´1 +i +ˆθnqp1 ` R´1 +i W J +i ˆθn ´ 2´1R´2 +i }ˆθn}2 ` δ1iq “ 0 , +(S.12) +where δ1i “ OptpR´1 +i W J +i ˆθn ´ 2´1R´2 +i }ˆθn}2q2u “ Oppn´1q. By Markov’s inequality, for any +ε ą 0, +P +ˆ +max +1ďiďn R´1 +i +ě εζ1n1{4 +˙ +“ +P +ˆ +max +1ďiďn R´4 +i +ě ε4ζ4 +1n +˙ +ď +E +ˆ +max +1ďiďn R´4 +i +˙ +{pε4ζ4 +1nq ď nEpR´4 +i q{pε4ζ4 +1nq À ε´4 , +where the last inequality is due to Condition C.2. +Thus, max1ďiďn R´2 +i +“ Oppζ2 +1n1{2q, and +consequently, max1ďiďn δ1i “ Opp}ˆθn}2 max1ďiďn R´2 +i q “ Oppn´1{2q. Rewrite (S.12) as +n´1 +nÿ +i“1 +p1 ´ 2´1R´2 +i }ˆθn}2 ` δ1iqWi ` n´1 +nÿ +i“1 +R´1 +i pW J +i ˆθnqWi +“ +n´1 +nÿ +i“1 +R´1 +i p1 ´ 2´1R´2 +i }ˆθn}2 ` δ1iqˆθn ` n´1 +nÿ +i“1 +R´2 +i pW J +i ˆθnqˆθn , +which implies +n´1 +nÿ +i“1 +p1 ´ 2´1R´2 +i }ˆθn}2 ` δ1iqWi ` n´1 +nÿ +i“1 +R´1 +i pW J +i ˆθnqWi +(S.13) +“ +n´1 +nÿ +i“1 +R´1 +i p1 ` δ1i ` δ2iqˆθn , +where δ2i “ R´1 +i W J +i ˆθn ´ 2´1R´2 +i }ˆθn}2 “ Oppδ1{2 +1i q satisfies max1ďiďn δ2i “ Oppn´1{4q. It is +30 + +straightforward to check that n´1 řn +i“1 R´1 +i pW J +i ˆθnqWi “ n´1 řn +i“1 R´1 +i WiW J +i ˆθn “ Qˆθn. From +Lemma A3, +|Qjℓ| À ζ1p´1|ωjℓ| ` Oppζ1n´1{2p´1 ` ζ1p´7{6 ` ζ1p´1´δ{2q, +and this implies that +|Qˆθn|8 ď }Q}1||ˆθn|8 À ζ1p´1}Ω}1|ˆθn|8 ` Oppζ1n´1{2 ` ζ1p´1{6 ` ζ1p´δ{2q|ˆθn|8. +According to Lemma A4, we have that |n´1 řn +i“1 ζ´1 +1 Wi|8 “ Optn´1{2 log1{2pnpqu. Then, +ˇˇˇˇˇζ´1 +1 n´1 +nÿ +i“1 +δ1iWi +ˇˇˇˇˇ +2 +8 +ď +ˇˇˇˇˇn´1 +nÿ +i“1 +pζ´1 +1 Wiq2 +ˇˇˇˇˇ +8 +˜ +n´1 +nÿ +i“1 +δ2 +1i +¸ +À Oppn´2q. +In addition, we have that |ζ´1 +1 n´1 řn +i“1 R´2 +i }ˆθn}2Wi|8 À Oppn´1q. Regarding equation (S.13) +and the fact that ζ´1 +1 n´1 řn +i“1 R´1 +i +“ 1 ` Oppn´1{2q , we obtain +ˆθn|8 +À +ˇˇˇˇˇζ´1 +1 n´1 +nÿ +i“1 +Wi +ˇˇˇˇˇ +8 +` ζ´1 +1 |Qˆθn|8 +À +p´1a0ppq|ˆθn|8 ` Oppn´1{2 ` p´p1{6^δ{2qq|ˆθn|8 ` Optn´1{2 log1{2pnpqu . +Thus, we conclude that |ˆθn|8 “ Optn´1{2 log1{2pnpqu as a0ppq — p1´δ. In addition, we have +|ζ´1 +1 Qˆθn|8 “ Optn´1{2p´p1{6^δ{2q log1{2pnpq`n´1 log1{2pnpqu and n´1 řn +i“1 R´1 +i p1`δ1i `δ2iq “ +ζ1t1 ` Oppn´1{4qu. Finally, we can write +n1{2pˆθn ´ θq “ n´1{2ζ´1 +1 +nÿ +i“1 +Wi ` Cn , +where Cn satisfies |Cn|8 “ Optn´1{4 log1{2pnpq ` p´p1{6^δ{2q log1{2pnpqu. +Proof of Theorem 1. Let Ln,p “ n´1{4 log1{2pnpq`p´p1{6^δ{2q log1{2pnpq. Then, for any sequence +ηn Ñ 8 and any t P Rp, +P +! +n1{2pˆθn ´ θq ď t +) +“ P +˜ +n´1{2ζ´1 +1 +nÿ +i“1 +Wi ` Cn ď t +¸ +31 + +ď +P +˜ +n´1{2ζ´1 +1 +nÿ +i“1 +Wi ď t ` ηnLn,p +¸ +` Pp|Cn|8 ą ηnLn,pq . +According to Lemma A4, Etpζ´1 +1 Wi,jq4u À ¯ +M2, Etpζ´1 +1 Wi,jq2u Á m, and }ζ´1 +1 Wi,j}ψα À ¯B for +all i “ 1, . . . , n and j “ 1, . . . , p. According to the Gaussian approximation for independent +partial sums in Koike (2021), let G „ Np0, ζ´2 +1 Bq with B “ EpW1W J +1 q, we have +P +˜ +n´1{2ζ´1 +1 +nÿ +i“1 +Wi ď t ` ηnLn,p +¸ +ď PpG ď t ` ηnLnpq ` O +´␣ +n´1 log5pnpq +(1{6¯ +ď +PpG ď tq ` OtηnLnp log1{2ppqu ` O +´␣ +n´1 log5pnpq +(1{6¯ +, +where the last inequality is from Nazarov’s inequality in Lemma A6. It is also worth noting that +the order O +´␣ +n´1 log5pnpq +(1{6¯ +is improved to O +´␣ +n´1 log5pnpq +(1{4¯ +in Chernozhukov et al. +(2019). Thus, +P +! +n1{2pˆθn ´ θq ď t +) +ď +PpG ď tq ` OtηnLnp log1{2ppqu ` O +´␣ +n´1 log5pnpq +(1{6¯ +`Pp|Cn|8 ą ηnLn,pq . +On the other hand, we also have +P +! +n1{2pˆθn ´ θq ď t +) +ě +PpG ď tq ´ OtηnLnp log1{2ppqu ´ O +´␣ +n´1 log5pnpq +(1{6¯ +´Pp|Cn|8 ą ηnLn,pq , +where Pp|Cn|8 ą ηnLn,pq Ñ 0 as n Ñ 8 according to Lemma 1. +Then, if log p “ opn1{5q and log n “ opp1{3^δq, with sufficiently slow ηn Ñ 8, we have +sup +tPRp +ˇˇˇPtn1{2pˆθn ´ θq ď tu ´ PpG ď tq +ˇˇˇ Ñ 0 . +32 + +We obtain immediately from Corollary 5.1 in Chernozhukov, Chetverikov and Kato (2017) that +ρnpAreq “ sup +APAre +ˇˇˇPtn1{2pˆθn ´ θq P Au ´ P pG P Aq +ˇˇˇ Ñ 0 , +which leads to the conclusion of this theorem. +Proof of Theorem 2. Let ˜Xi “ Xi ´ ˆθn and ˜Ri “ } ˜Xi} for i “ 1, . . . , n. According to Lemma +A5, +n1{2˜θn “ n´1{2ζ´1 +1 +nÿ +i“1 +ZiWi ` ˜Cn , +where ˜Cn satisfies | ˜Cn|8 “ Optn´1{4 log1{2pnpq ` p´p1{6^δ{2q log1{2pnpqu. +Denote ¯Wn “ n´1 řn +i“1 Wi and rewrite +n1{2˜θn “ n´1{2ζ´1 +1 +nÿ +i“1 +ZipWi ´ ¯Wnq ` +˜ +n´1{2ζ´1 +1 +nÿ +i“1 +Zi +¸ +¯Wn ` ˜Cn, +where +ˇˇˇˇˇ +˜ +n´1{2ζ´1 +1 +nÿ +i“1 +Zi +¸ +¯Wn +ˇˇˇˇˇ +8 +ď ζ´1 +1 +ˇˇˇˇˇn´1{2 +nÿ +i“1 +Zi +ˇˇˇˇˇ +ˇˇ ¯Wn +ˇˇ +8 À n´1{2 log1{2pnpq +according to Lemma A4 (iii). +It is clear that E˚ ␣ +n´1{2ζ´1 +1 +řn +i“1 ZipWi ´ ¯Wnq +( +“ 0. Let ˆB “ n´1 řn +i“1 WiW J +i , then +Var˚ +# +n´1{2ζ´1 +1 +nÿ +i“1 +ZipWi ´ ¯Wnq ++ +“ ζ´1 +1 +ˆB ´ ζ´2 +1 +¯Wn ¯W J +n . +Denote Bjℓ and ˆBjℓ be the pj, ℓqth element of B and ˆB, respectively. In addition, denote ¯Wn,j +as the jth element of ¯Wn. Define +∆n “ max +1ďj,ℓďp +ˇˇˇζ´2 +1 +ˆBjℓ ´ ζ´2 +1 +¯Wn,j ¯Wn,ℓ ´ ζ´2 +1 Bjℓ +ˇˇˇ , +33 + +then +∆n +ď +∆n1 ` max +1ďj,ℓďp +ˇˇζ´2 +1 +¯Wn,j ¯Wn,ℓ +ˇˇ À ∆n1 ` n´1 logpnpq, +where +∆n1 +“ +max +1ďj,ℓďp |ζ´2 +1 +ˆBjℓ ´ ζ´2 +1 Bjℓ| “ max +1ďj,ℓďp +ˇˇˇˇˇn´1ζ´2 +1 +nÿ +i“1 +tWi,jWi,ℓ ´ EpWi,jWi,ℓqu +ˇˇˇˇˇ . +From the properties of the ψα norm, it holds that +›››› +max +1ďiďn;1ďj,ℓďp |ζ´2 +1 Wi,jWi,ℓ| +›››› +ψα{2 +À +›››› +max +1ďiďn,1ďjďp |ζ´2 +1 Wi,j|2 +›››› +ψα{2 +“ +ζ´2 +1 +›››› +max +1ďiďn,1ďjďp |Wi,j| +›››› +2 +ψα +À log2pnpq . +Let Jn “ max1ďiďn;1ďj,ℓďp ζ´2 +1 |Wi,jWi,ℓ ´ EpWi,jWi,ℓq|, and +σ2 +n +“ +max +1ďj,ℓďp ζ´2 +1 +nÿ +i“1 +EtWi,jWi,ℓ ´ EpWi,jWi,ℓqu2 +À +max +1ďj,ℓďp ζ´2 +1 +nÿ +i“1 +Et|Wi,jWi,ℓ|2u À n . +It also follows that +}Jn}ψα{2 À ζ´2 +1 +›››› +max +1ďiďn;1ďj,ℓďp |Wij,Wi,ℓ| +›››› +ψα{2 +` +max +1ďiďn;1ďj,ℓďp ζ´2 +1 Ep|Wi,jWi,ℓ|q À log2pnpq . +By Lemma E.1 in Chernozhukov, Chetverikov and Kato (2017), it holds that +Ep∆n1q +À +n´1 ” +σn log1{2ppq ` tEpJ2 +nqu1{2 log p +ı +À +n´1 ! +n1{2 log1{2ppq ` log1{α`1pnpq +) +À +n´1{2 log1{2pnpq . +Then applying Lemma E.2 in Chernozhukov, Chetverikov and Kato (2017) with η “ 1 and +34 + +β “ α{2, we obtain that +Pp∆n1 ě 2Ep∆nq ` tq À exp +` +´C1nt2˘ +` 3 exp +! +´C2ttn log´2{αpnpquα{2) +. +Thus, there exist a constant C1 depends on δ such that +P +! +∆n1 ą C1n´1{2 log1{2pnpq +) +À p´δ Ñ 0 . +From the multiplier bootstrap theorem and Gaussian comparison in Chernozhukov, Chetverikov +and Kato (2017) and Koike (2021), +sup +tPRp +ˇˇˇˇˇP˚ +# +n´1{2ζ´1 +1 +nÿ +i“1 +ZipWi ´ ¯Wnq ď t ++ +´ PpG ď tq +ˇˇˇˇˇ +À +∆1{2 +n +logppq ` tn´1 log5pnpqu1{4 , +on t∆n À n´1{2 log1{2pnpqu, which occurs with probability 1 ´ p´δ. +Finally, similar to the proof of Theorem 1, we can show that under Conditions C.2 and C.3 +with a0ppq — p1´δ, if log p “ opn1{5q and log n “ opp1{3^δq, we have +sup +APAre +ˇˇˇPtn1{2pˆθn ´ θq P Au ´ P˚pn1{2˜θn P Aq +ˇˇˇ Ñ 0 +in probability, which completes the proof of this theorem. +Proof of Theorem 3. Theorems 1 and 2 indicates that there exists a positive sequence βn,p Ñ 0 +as n, p Ñ 8 such that +sup +tPR +ˇˇˇPpn1{2|ˆθn ´ θ|8 ď tq ´ Pp|G|8 ď tq +ˇˇˇ ď βn,p{2 +and +sup +tPR +ˇˇˇPpn1{2|ˆθn ´ θ|8 ď tq ´ P˚pn1{2|˜θn|8 ď tq +ˇˇˇ ď βn,p +with probability approaching one when n Ñ 8. +Letting q1´α be the p1 ´ αqth quantile of +35 + +n1{2|ˆθn ´ θ|8, that is, q1´α “ inftu P R : Ppn1{2|ˆθn ´ θ|8 ą uq ď αu. Then, +P˚pn1{2|˜θn|8 ď q1´α`βn,pq ě Ppn1{2|ˆθn ´ θ|8 ď q1´α`βn,pq ´ βn,p ě 1 ´ α , +with probability approaching one as n Ñ 8. +On the other hand, it holds with the same +probability that +P˚pn1{2|˜θn|8 ď q1´α´3βn,pq +ď +Ppn1{2|ˆθn ´ θ|8 ď q1´α´3βn,pq ` βn,p +“ +Ppn1{2|ˆθn ´ θ|8 ď q1´α´3βn,p ´ n´1{6q ` βn,p +`Ppn1{2|ˆθn ´ θ|8 ď q1´α´3βn,pq +´Ppn1{2|ˆθn ´ θ|8 ď q1´α´3βn,p ´ n´1{6q +ă +1 ´ α ´ 2βn,p ` Ppn1{2|ˆθn ´ θ|8 ď q1´α´3βn,pq +´Ppn1{2|ˆθn ´ θ|8 ď q1´α´3βn,p ´ n´1{6q , +where Ppn1{2|ˆθn ´ θ|8 ď q1´α´3βn,pq ´ Ppn1{2|ˆθn ´ θ|8 ď q1´α´3βn,p ´ n´1{6q can be bounded +by +ˇˇˇPpn1{2|ˆθn ´ θ|8 ď q1´α´3βn,pq +´Ppn1{2|ˆθn ´ θ|8 ď q1´α´3βn,p ´ n´1{6q +ˇˇˇ +ď +ˇˇˇPp|G|8 ď q1´α´3βn,pq ´ Pp|G|8 ď q1´α´3βn,p ´ n´1{6q +ˇˇˇ +` +ˇˇˇPpn1{2|ˆθn ´ θ|8 ď q1´α´3βn,pq ´ Pp|G|8 ď q1´α´3βn,pq +ˇˇˇ +` +ˇˇˇPpn1{2|ˆθn ´ θ|8 ď q1´α´3βn,p ´ n´1{6q +´ Pp|G|8 ď q1´α´3βn,p ´ n´1{6q +ˇˇˇ +ď +ˇˇˇPp|G|8 ď q1´α´3βn,pq ´ Pp|G|8 ď q1´α´3βn,p ´ n´1{6q +ˇˇˇ ` βn,p +ď +C1 +␣ +n´1 log5pnpq +(1{6 ` βn,p, +for some positive constant C1, where the last inequality follows from the Nazarov’s inequality. +36 + +Choosing C1 +␣ +n´1 log5pnpq +(1{6 ď βn,p, we obtain +P˚pn1{2|˜θn|8 ď q1´α´3βn,pq ă 1 ´ α +with probability approaching one. It follows that +Ppq1´α´3βn,p ă qB +1´α ď q1´α`βn,pq Ñ 1, +as n, p Ñ 8 . +Therefore, +Ppn1{2|ˆθn ´ θ|8 ą qB +1´αq +ď +Ppn1{2|ˆθn ´ θ|8 ą q1´α´3βn,pq ` PpqB +1´α ď q1´α´3βn,pq +ď +α ` 3βn,p ` op1q +(S.14) +and +Ppn1{2|ˆθn ´ θ|8 ą qB +1´αq +ě +Ppn1{2|ˆθn ´ θ|8 ą q1´α`βn,pq ´ PpqB +1´α ą q1´α`βn,pq +ě +Ppn1{2|ˆθn ´ θ|8 ą q1´α`βn,p ´ n´1{6q ´ op1q +`Ppn1{2|ˆθn ´ θ|8 ą q1´α`βn,pq +´Ppn1{2|ˆθn ´ θ|8 ą q1´α`βn,p ´ n´1{6q +ě +α ´ 2βn,p ´ C2 +␣ +n´1 log5pnpq +(1{6 ě α ´ 3βn,p. +for some positive constant C2, where the second last inequality follows from the Nazarov’s +inequality and the last inequality is from choosing βn,p ě C2 +␣ +n´1 log5pnpq +(1{6. +Finally, as +βn,p Ñ 0, +Ppn1{2|ˆθn ´ θ|8 ě qB +1´αq ´ α Ñ 0, +which completes the proof of this theorem. +Proof of Theorem 4. Without loss of generality, we assume θ0 “ 0. Rewrite the test statistic as +Tn “ n1{2|ˆθn|8, and let T c +n “ n1{2|ˆθn ´ θ|8, which has the same distribution of Tn under H0. +37 + +Then, it holds that +Tn ě n1{2|θ|8 ´ T c +n . +Therefore, the power of the test based on Tn satisfies +PpTn ą qB +1´α | H1q +ě +Ppn1{2|θ|8 ´ T c +n ě qB +1´α | H1q +“ +PpT c +n ď n1{2|θ|8 ´ qB +1´α | H1q +Under the conditions of Theorem 2, there exists a positive sequence βn,p Ñ 0 as n, p Ñ 8, +satisfies +sup +tPR +|PpT c +n ą t | H1q ´ Pp|G|8 ą t | H1q| ď βn,p, +(S.15) +where G „ Np0, ζ´2 +1 Bq. Letting q1´α be the p1 ´ αqth quantile of T C +n and qG +1´α be the p1 ´ αqth +quantile of |G|8. Choosing t “ qG +1´α`2βn,p in equation (S.15), we obtain that |PpT c +n ą qG +1´α`2βn,p | +H1q ´ α ` 2βn,p| ď βn,p and PpT c +n ą qG +1´α`2βn,p | H1q ď α ´ βn,p, which implies that q1´α`βn,p ď +qG +1´α`2βn,p. +Note that qB +1´α is the p1´αqth quantile of n1{2|˜θn|8 conditional on X1, . . . , Xn. By carrying +out similar procedure as in the proof of equation (S.14), we can show that +PpT c +n ą n1{2|θ|8 ´ qB +1´α | H1q ď PpT c +n ą n1{2|θ|8 ´ q1´α`βn,p | H1q ` op1q . +(S.16) +It follows that +PpT c +n ą n1{2|θ|8 ´ qB +1´α | H1q ď PpT c +n ą n1{2|θ|8 ´ qG +1´α`2βn,p | H1q ` op1q . +For |G|8, we know that }|G|8}ψ2 À log1{2pnpq. In addition, for any t ą 0, +Pp|G|8 ą tq ď 2 expt´C1pt{}|G|8}ψ2q2u ď 2 expt´C2t2 log´1pnpqu . +Choosing t “ C´1{2 +2 +log1{2p2{pα ´ 2βn,pqq log1{2pnpq, we arrive at +Pp|G|8 ą C´1{2 +2 +log1{2p2{pα ´ 2βn,pqq log1{2pnpqq ď α ´ 2βn,p , +38 + +which leads to +qG +1´α`2βn,p ď C´1{2 +2 +log1{2p2{pα ´ 2βn,pqq log1{2pnpq. +Then, if |θ|8 ě Cn´1{2 log1{2pα´1q log1{2pnpq for a large enough constant C, it holds with +sufficiently large C3 that +PpTn ą qB +1´α | H1q +ě +PpT c +n ď n1{2|θ|8 ´ qG +1´α`2βn,p | H1q ` op1q +ě +PtT c +n ď C3 log1{2pnpq log1{2pα´1q | H1u ` op1q +ě +Pt|G|8 ď C3 log1{2pnpq log1{2pα´1q | H1u ´ βn,p ` op1q +ě +1 ´ 2αC2C2 +3 ´ βn,p ` op1q. +We complete the proof of this theorem. +Proof of Theorem 5. Recall that ˆζ1 “ n´1 řn +i“1 }Xi ´ ˆθn}´1. It has been shown in the proof of +Lemma A5 that +}Xi ´ ˆθn}´1 “ R´1 +i +´ +1 ` R´1 +i W J +i ˆθn ´ 2´1R´2 +i }ˆθn}2 ` ˜δ1i +¯ +, +where ˜δ1i satisfies ˜δ1i “ Oppn´1q and max1ďiďn ˜δ1i “ Oppn´1{2q. Thus, +ˆζ1 +“ +n´1 +nÿ +i“1 +R´1 +i +´ +1 ` R´1 +i W J +i ˆθn ´ 2´1R´2 +i }ˆθn}2 ` ˜δ1i +¯ +“ +n´1 +nÿ +i“1 +R´1 +i p1 ` ˜δ3iq, +where ˜δ3i satisfies ˜δ3i “ Oppn´1{2q and max1ďiďn ˜δ3i “ Oppn´1{4q. By the fact that n´1 řn +i“1 R´1 +i +“ +ζ1 ` Oppζ1n´1{2q, we conclude that +ˆζ1{ζ1 ´ 1 “ Oppn´1{2q. +39 + +Let ˜Wi “ pXi ´ ˆθnq{}Xi ´ ˆθn} for i “ 1, . . . , n. From the proof of Lemma A5, +˜Wi “ pWi ´ R´1 +i +ˆθnqp1 ` ˜δ2iq “ Wi ` Wi˜δ2i ´ R´1 +i +ˆθnp1 ` ˜δ2iq, +where ˜δ2i satisfies ˜δ2i “ Oppn´1{2q and max1ďiďn ˜δ2i “ Oppn´1{4q. Let ˜Wi,j be the jth component +of ˜Wi, then +ˆBjj +“ +n´1 +nÿ +i“1 +˜W 2 +i,j +“ +n´1 +nÿ +i“1 +W 2 +i,jt1 ` Opp˜δ2iqu ` n´1 +nÿ +i“1 +R´1 +i Wi,j ˆθn,jt1 ` Opp˜δ2iqu +`n´1 +nÿ +i“1 +R´2 +i +ˆθ2 +n,jt1 ` Opp˜δ2iqu, +where max1ďjďp n´1 řn +i“1 W 2 +i,jt1 ` Opp˜δ2iqu{Bjj “ 1 ` Oppn´1{4q, +max +1ďjďp +ˇˇˇˇˇn´1 +nÿ +i“1 +R´2 +i +ˆθ2 +n,jt1 ` Opp˜δ2iqu +››››› +À +ˇˇˇˇˇn´1 +nÿ +i“1 +R´2 +i +ˇˇˇˇˇ max +1ďjďp |ˆθ2 +n,j| +“ +Optζ2 +1n´1 log1{2pnpqu +and +max +1ďjďp +ˇˇˇˇˇn´1 +nÿ +i“1 +R´1 +i Wi,j ˆθn,jt1 ` Opp˜δ2iqu +ˇˇˇˇˇ +À +˜ +n´1 +nÿ +i“1 +R´2 +i +¸1{2 +max +1ďjďp +$ +& +% +˜ +n´1 +nÿ +i“1 +W 2 +i,j +¸1{2, +. +- max +1ďjďp |ˆθn,j| +“ +Optζ1p´1{2n´1{2 log1{2pnpqu. +It follows that +max +1ďjďp +ˆBjj{Bjj “ 1 ` Optn´1{4 log1{2pnpqu. +40 + +Thus, Condition A (ii) of Belloni et.al (2018) is satisfied by sn,j. It is clear that Condition A +(i) of Belloni et.al (2018) is satisfied by the remainder term Cn. Hence, from Theorem 2.4 in +Belloni et.al (2018), for any 1 ď j ď p, if log p “ opn1{5q and log n “ opp1{3^δq, we have +sup +0ďxď21{2 log1{2pnpq +ˇˇˇP +! +n1{2pˆθn,j ´ θjq{sn,j ą x +) +´ t1 ´ Φpxqu +ˇˇˇ Ñ 0 . +Let ¯Tj “ n´1{2 řn +i“1 Wi,j{tn´1 řn +i“1 W 2 +i,j ´ pn´1 řn +i“1 Wi,jq2u1{2. Based on Equation (13) of +Liu and Shao (2014), for any sequence dn Ñ 8 and dn “ oppq as n Ñ 8, with Condition C.4, +sup +0ďtďG´1 +κ pdn{pq +ˇˇˇˇˇ +ř +jPH0 It| ¯Tj| ě tu +p0Gκptq +´ 1 +ˇˇˇˇˇ “ opp1q , +where Gκptq is some function such that Gκptq ě Gptq “ 2t1 ´ Φptqu for all t P R, and Gκptq “ +Gptqt1 ` op1qu uniformly over 0 ď t ď 21{2 log1{2ppq. Then, with enough large n, as long as +|H| Ñ 8 and |H| ą 2{α, we have +α|H|{p ě 2{p “ 2 expt´p21{2 log1{2 pq2{2u ě 2t1 ´ Φp21{2 log1{2 pqu “ Gp21{2 log1{2 pq . +It follows that G´1pα|H|{pq ď 21{2 log1{2 p, and consequently, +sup +0ďtďG´1pα|H|{pq +ˇˇˇˇ +ř +jPH0 It| ¯Tj| ě tu +p0Gptq +´ 1 +ˇˇˇˇ “ opp1q . +Let T 1 +j “ n1{2pˆθn,j ´ θjq{sn,j, we obtain max1ďjďp |T 1 +j ´ ¯Tj| “ optlog´1{2ppqu with some careful +calculations. With similar procedure to Page 84 of Belloni et.al (2018), it holds that +sup +0ďtďG´1pα|H|{pq +ˇˇˇˇ +ř +jPH0 It|T 1 +j| ě tu +p0Gptq +´ 1 +ˇˇˇˇ “ opp1q . +(S.17) +The B-H method with P1, . . . , Pp is equivalent to the following procedure: reject H0j, if only +if Pj ď ˆt0, where +ˆt0 “ sup +" +0 ď t ď 1 : t ď +α maxtřp +j“1 IpPj ď tqu +p +* +. +Then we have ˆt0 “ +α maxtřp +j“1 IpPjďˆt0q,1u +p +, and α|H|{p ě Gp21{2 log1{2 pq. Set t “ G´1pα|H|{pq, +41 + +then t ď 21{2 log1{2 p with probability tends to 1. Thus, we have +Gptq “ α|H| +p +ď +α maxtřp +j“1 Ip|Ti| ě 21{2 log1{2 pq, 1u +p +ď +α maxtřp +j“1 Ip|Ti| ě tq, 1u +p +, +where the second inequality implied by (B.29) of Belloni et.al (2018). It implies that Ppˆt0 ě +α|H|{pq Ñ 1 with ˆt0 “ Gpˆtq, and together with (S.17), we have +ř +jPH0 It|T 1 +j| ě ˆtu +p0Gpˆtq +“ +ř +jPH0 Ip|Tj| ě ˆtq +p0Gpˆtq +Ñ 1 , +which is equivalent to +ř +jPH0 IpPj ď ˆt0q +p0ˆt0 +Ñ 1 . +Finally, +FDRM “ +ř +jPH0 IpPj ď ˆt0q +maxtřp +j“1 IpPj ď ˆt0q, 1u “ +ř +jPH0 IpPj ď ˆt0q +pˆt0{α +Ñ αp0 +p +as n Ñ 8, which completes the proof of this theorem. +Appendix B: Proof of preliminary lemmas +In this section, we present proofs of preliminary lemmas in Section A1 of Appendix A. +Proof of Lemma A1. As the components of U1 are independent and standardized, simple cal- +culations yield Ep}U1}2q “ p and +Ep}ΓU1}2q “ EpU J +1 ΓJΓU1q “ trtΓJΓEpU1U J +1 qu “ trpΩq . +Under Condition C.1, the components of U1 “ pU1,1, . . . , U1,pqJ are independent sub-exponential +random variables such that max1ďjďp }U1,j}ψα ď c0. Applying the concentration inequality in +the proof of Lemma S2.1 in (Wang, Peng and Li 2015), for every t ě 0, +P +`ˇˇ}U1}2 ´ p +ˇˇ ě t +˘ +ď C1 exp +! +´C2 +` +p´1t2˘α{p4α`4q) +. +(S.18) +42 + +and +P +␣ˇˇ}ΓU1}2 ´ trpΩq +ˇˇ ě t +( +ď C1 exp +„ +´C2 +! +t2 +trpΩ2q +)α{p4α`4qȷ +. +(S.19) +For any fixed 0 ă ϵ ă 1, let +A1 “ tp ´ ϵpp1`δq{2 ď }U1}2 ď p ` ϵpp1`δq{2u +and +A2 “ tp1 ´ ϵqtrpΩq ď }ΓU1}2 ď p1 ` ϵqtrpΩqu. +Taking t “ ϵpp1`δq{2 in (S.18) and t “ ϵtrpΩq in (S.19), we have +PpA1q ě 1 ´ C1 exp +! +´C2pϵ2pδqα{p4α`4q) +and +PpA2q ě 1 ´ C1 exp +« +´C2 +"ϵ2tr2pΩq +trpΩ2q +*α{p4α`4qff +. +Under Condition C.3, +trpΩ2q “ +pÿ +j“1 +pÿ +ℓ“1 +ω2 +jℓ ď ¯ +Mp max +1ďℓďp +pÿ +j“1 +|ωjℓ| ď ¯ +Mpa0ppq . +Since trpΩq “ p and a0ppq — p1´δ, we conclude that +tr2pΩq +trpΩ2q ě +p2 +¯ +Mpa0ppq — pδ . +Consequently, for some positive constants c1 and c2, we get that +PpA1q ě 1 ´ c1 expt´c2pδα{p4α`4qu . +43 + +and +PpA2q ě 1 ´ c1 expt´c2pδα{p4α`4qu +for sufficient large p. Thus, we finish the proof of this lemma. +Proof of Lemma A2. (i) As the components of Ui “ pUi,1, . . . , Ui,pqJ are i.i.d. standardized sub- +exponential random variables, simple algebra yields +Ep}Ui}4q +“ +E +$ +& +% +˜ pÿ +j“1 +U 2 +i,j +¸2, +. +- +“ +pÿ +j“1 +EpU 4 +i,jq ` +ÿ +1ďj1‰j2ďp +EpU 2 +i,j1qEpU 2 +i,j2q +“ +pEpU 4 +i,jq ` ppp ´ 1q +and +Ep}Ui}6q +“ +pÿ +j“1 +EpU 6 +i,jq ` 3 +ÿ +1ďj1‰j2ďp +EpU 4 +i,j1qEpU 2 +i,j2q +` +ÿ +1ďj1‰j2‰j3ďp +EpU 2 +i,j1qEpU 2 +i,j2qEpU 2 +i,j3q +“ +pEpU 6 +i,jq ` 3ppp ´ 1qEpU 4 +i,jq ` ppp ´ 1qpp ´ 2q . +In addition, +Ep}Ui}8q +“ +pÿ +j“1 +EpU 8 +i,jq ` 4 +ÿ +1ďj1‰j2ďp +EpU 6 +i,j1qEpU 2 +i,j2q +`3 +ÿ +1ďj1‰j2ďp +EpU 4 +i,j1qEpU 4 +i,j2q +`6 +ÿ +1ďj1‰j2‰j3ďp +EpU 4 +i,j1qEpU 2 +i,j2qEpU 2 +i,j3q +` +ÿ +1ďj1‰j2‰j3‰j4ďp +EpU 2 +i,j1qEpU 2 +i,j2qEpU 2 +i,j3qEpU 2 +i,j4q +“ +pEpU 8 +i,jq ` 4ppp ´ 1qEpU 6 +i,j1q ` 3ppp ´ 1qtEpU 4 +i,j1qu2 +`3ppp ´ 1qEpU 4 +i,jq ` ppp ´ 1qpp ´ 2qpp ´ 3q . +44 + +The result of Ep}Ui}2kq “ pk ` Oppk´1q for any positive integer k can be checked by +Ep}Ui}2kq +“ +ÿ +1ďj1‰¨¨¨‰jkďp +EpU 2 +i,j1q ˆ ¨ ¨ ¨ ˆ EpU 2 +i,jkqt1 ` Opp´1qu +“ +pk ` Oppk´1q . +Moreover, by the fact that t1 ` u ´ pu ´ 1q2u{2 ď u1{2 ď p1 ` uq{2 for all u ě 0, we can get that +Ep}U}kq “ pk{2 ` Oppk{2´1q for all positive integer k. +(ii) Write Λjℓ “ řp +j1“1 Γj1jΓj1ℓ as the pj, ℓqth element of ΓJΓ, then +Ep}ΓUi}4q +“ +E +$ +& +% +˜ pÿ +j“1 +pÿ +ℓ“1 +ΛjℓUi,jUi,ℓ +¸2, +. +- +“ +pÿ +j“1 +Λ2 +jjEpU 4 +i,jq ` 2 +ÿ +1ďj1‰j2ďp +Λ2 +j1j2EpU 2 +i,j1qEpU 2 +i,j2q +` +ÿ +1ďj1‰j2ďp +Λj1j1Λj2j2EpU 2 +i,j1qEpU 2 +i,j2q +“ +EpU 4 +i,jq +pÿ +j“1 +Λ2 +jj ` 2 +ÿ +1ďj1‰j2ďp +Λ2 +j1j2 ` +ÿ +1ďj1‰j2ďp +Λj1j1Λj2j2 +“ +˜ pÿ +j“1 +Λjj +¸2 +` tEpU 4 +i,jq ´ 1u +pÿ +j“1 +Λ2 +jj ` 2 +ÿ +1ďj1‰j2ďp +Λ2 +j1j2 +“ +ttrpΩqu2 ` OttrpΩ2qu +as řp +j“1 Λ2 +jj`ř +1ďj1‰j2ďp Λ2 +j1j2 “ řp +j“1 +řp +ℓ“1 Λ2 +jℓ “ trpΩ2q and trpΩ2q À p2´δ based on Condition +C.3. Similarly, we can show that +Ep}ΓUi}6q +“ +ÿ +1ďj1‰j2‰j3ďp +pΛj1j1Λj2j2Λj3j3 ` Λ2 +j1j2Λj3j3 +` Λj1j2Λj1j3Λj2j3qEpU 2 +i,j1qEpU 2 +i,j2qEpU 2 +i,j3qt1 ` Opp´1qu +“ +p3 ` Opp3´δq +and Ep}ΓUi}12q “ p6 ` Opp6´δq. +Similar to the proof of part (i), the result Ep}ΓUi}q “ p1{2`Opp1{2´δq and Ep}ΓUi}3q “ p3{2` +Opp3{2´δq are directly consequences of Ep}ΓUi}2q “ p, Ep}ΓUi}4q “ p2 ` Opp2´δq, Ep}ΓUi}6q “ +45 + +p3 ` Opp3´δq, Ep}ΓUi}12q “ p6 ` Opp6´δq and t1 ` u ´ pu ´ 1q2u{2 ď u1{2 ď p1 ` uq{2 for all +u ě 0. +(iii) Now we consider Et}ΓSpUiq}2u. For i “ 1, . . . , n, let +A1i “ tp ´ ϵpp1`δq{2 ď }Ui}2 ď p ` ϵpp1`δq{2u +for a fixed 0 ă ϵ ă 1. According to Lemma A1 and the fact that }ΓUi}2 ď trpΩq}Ui}2, +Et}ΓSpUiq}2u +“ +Ep}ΓUi}2}Ui}´2q +“ +p´1Et}ΓUi}2u ` E +␣ +}ΓUi}2 ` +}Ui}´2 ´ p´1˘( +“ +1 ` E +␣ +}ΓUi}2 ` +}Ui}´2 ´ p´1˘( +, +where +E +␣ +}ΓUi}2 ` +}Ui}´2 ´ p´1˘( +ď +p´1E +` +}ΓUi}2}Ui}´2 ˇˇ}Ui}2 ´ p +ˇˇ˘ +“ +p´1E +␣ +}ΓUi}2}Ui}´2 ˇˇ}Ui}2 ´ p +ˇˇ IpA1iq +( +`p´1E +␣ +}ΓUi}2}Ui}´2 ˇˇ}Ui}2 ´ p +ˇˇ IpAc +1iq +( +ď +p´1tp ´ ϵpp1`δq{2u´1E +` +}ΓUi}2 ˇˇ}Ui}2 ´ p +ˇˇ˘ +`p´1trpΩqE +␣ˇˇ}Ui}2 ´ p +ˇˇ IpAc +1iq +( +ď +p´1tp ´ ϵpp1`δq{2u´1 ␣ +Ep}ΓUi}4q +(1{2 ! +Ep +ˇˇ}Ui}2 ´ p +ˇˇ2q +)1{2 +` +! +Ep +ˇˇ}Ui}2 ´ p +ˇˇ2q +)1{2 +tPpAc +1iqu1{2 +ď +p´1pp ´ ϵp1´δq´1tp2 ` Opp2´δqu1{2 ˆ Opp1{2q +`Opp1{2q ˆ c1{2 +1 +expt´c2pδα{p4α`4q{2u +“ +Opp´1{2q . +It follows that Et}ΓSpUiq}2u “ 1 ` Opp´1{2q. +46 + +Similarly, the last result follows from +Et}ΓSpUiq}4u +“ +p´2Et}ΓUi}4u ` E +␣ +}ΓUi}4 ` +}Ui}´4 ´ p´2˘( +“ +1 ` Opp´δq ` E +␣ +}ΓUi}4 ` +}Ui}´4 ´ p´2˘( +, +where +E +␣ +}ΓUi}4 ` +}Ui}´4 ´ p´2˘( +ď +p´2E +` +}ΓUi}4}Ui}´4 ˇˇ}Ui}4 ´ p2ˇˇ˘ +“ +p´2E +␣ +}ΓUi}4}Ui}´4 ˇˇ}Ui}4 ´ p2ˇˇ IpA1iq +( +`p´2E +␣ +}ΓUi}4}Ui}´4 ˇˇ}Ui}4 ´ p2ˇˇ IpAc +1iq +( +ď +p´2pp ´ ϵp1´δq´2E +` +}ΓUi}4 ˇˇ}Ui}4 ´ p2ˇˇ˘ +`p´2ttrpΩqu2E +␣ˇˇ}Ui}4 ´ p2ˇˇ IpAc +1iq +( +ď +p´2pp ´ ϵp1´δq´2 ␣ +Ep}ΓUi}6q +(2{3 ! +Ep +ˇˇ}Ui}4 ´ p2ˇˇ3q +)1{3 +` +! +Ep +ˇˇ}Ui}4 ´ p2ˇˇ2q +)1{2 +tPpAc +1iqu1{2 +ď +p´2pp ´ ϵp1´δq´2 ˆ Opp2q ˆ Opp3{2q +`Opp3{2q ˆ c1{2 +1 +expt´c2pδα{p4α`4q{2u +“ +Opp´1{3q . +(iv) as νi and SpUiq are independent, +Epν´1 +i +qEt}ΓSpUiq}´1u +“ +Epν´1 +i +}ΓUi}´1}Ui}q +“ +EpR´1 +i }Ui}q +“ +EtR´1 +i }Ui}IpA1iqu ` EtR´1 +i }Ui}IpAc +1iqu +ď +tp ` ϵpp1`δq{2u1{2EtR´1 +i IpA1iqu ` tEpR´4 +i qu1{4tE}Ui}4u1{4tPpAc +1iqu1{2 +À +tp ` ϵpp1`δq{2u1{2EpR´1 +i q ` ζ1{4 +4 +ˆ p1{2 ˆ c1{2 +1 +expt´c2pδα{p4α`4q{2u +À +ζ1p1{2 , +47 + +and +Epν´2 +i +qEt}ΓSpUiq}´2u +“ +EtR´2 +i }Ui}2IpA1iqu ` EtR´2 +i }Ui}2IpAc +1iqu +ď +tp ` ϵpp1`δq{2uEtR´2 +i IpA1iqu ` tEpR´4 +i qu1{2tE}Ui}6u1{3tPpAc +1iqu1{6 +À +tp ` ϵpp1`δq{2uEpR´2 +i q ` ζ1{2 +4 +ˆ p ˆ c1{6 +1 +expt´c2pδα{p4α`4q{6u +À +ζ2p , +In addition, we also have +Epν´3 +i +qEt}ΓSpUiq}´3u +“ +EtR´3 +i }Ui}3IpA1iqu ` EtR´3 +i }Ui}3IpAc +1iqu +ď +tp ` ϵpp1`δq{2u3{2EtR´3 +i IpA1iqu ` tEpR´4 +i qu3{4tE}Ui}18u1{6tPpAc +1iqu1{12 +À +tp ` ϵpp1`δq{2u3{2EpR´3 +i q ` ζ3{4 +4 +ˆ p3{2 ˆ c1{12 +1 +expt´c2pδα{p4α`4q{12u +À +ζ3p3{2 . +By Cauchy-Schwarz inequality and Jensen’s inequality, we can show that +rEt}ΓSpUiq}´1us´1 ď Et}ΓSpUiq}u ď rEt}ΓSpUiq}2us1{2 “ 1 ` Opp´1{2q , +rEt}ΓSpUiq}´2us´1 ď Et}ΓSpUiq}2u “ 1 ` Opp´1{2q , +and +rEt}ΓSpUiq}´3us´1 ď Et}ΓSpUiq}3u ď rEt}ΓSpUiq}4us3{4 “ 1 ` Opp´1{3q . +Then, the results of this part follows immediately. We finish the proof of this lemma. +Proof of Lemma A3. (i) For i “ 1, . . . , n, let A2i “ tp1 ´ ϵqtrpΩq ď }ΓUi}2 ď p1 ` ϵqtrpΩqu for a +fixed 0 ă ϵ ă 1. Recall that Γj is the jth row of Γ and Wi,j “ ΓjUi{}ΓUi}, then +Qjℓ +“ +n´1 +nÿ +i“1 +R´1 +i Wi,jWi,ℓ “ n´1 +nÿ +i“1 +ν´1 +i +pΓjUiqpΓℓUiq}ΓUi}´3 +48 + +“ +n´1p´3{2 +nÿ +i“1 +ν´1 +i +pΓjUiqpΓℓUiq +`n´1 +nÿ +i“1 +ν´1 +i +pΓjUiqpΓℓUiq +´ +}ΓUi}´3 ´ p´2{3¯ +, +where the last term satisfies +ˇˇˇˇˇE +# +n´1 +nÿ +i“1 +ν´1 +i +pΓjUiqpΓℓUiq +´ +}ΓUi}´3 ´ p´2{3¯+ˇˇˇˇˇ +ď +p´3{2E +! +ν´1 +i +|pΓjUiqpΓℓUiq| }ΓUi}´3 ˇˇˇ}ΓUi}3 ´ p3{2ˇˇˇ +) +“ +p´3{2E +! +R´1 +i +|pΓjUiqpΓℓUiq| }ΓUi}´2 ˇˇˇ}ΓUi}3 ´ p3{2ˇˇˇ +) +“ +p´3{2E +! +R´1 +i +|pΓjUiqpΓℓUiq| }ΓUi}´2 ˇˇˇ}ΓUi}3 ´ p3{2ˇˇˇ IpA2iq +) +`p´3{2E +! +R´1 +i +|pΓjUiqpΓℓUiq| }ΓUi}´2 ˇˇˇ}ΓUi}3 ´ p3{2ˇˇˇ IpAc +2iq +) +ď +p1 ´ ϵq´1p´5{2E +! +R´1 +i +|pΓjUiqpΓℓUiq| +ˇˇˇ}ΓUi}3 ´ p3{2ˇˇˇ IpA2iq +) +`p´3{2E +! +R´1 +i +ˇˇˇ}ΓUi}3 ´ p3{2ˇˇˇ IpAc +2iq +) +À +p´5{2tEpR´4 +i qu1{4 ” +E +! +|pΓjUiqpΓℓUiq|4)ı1{4 " +E +ˆˇˇˇ}ΓUi}3 ´ p3{2ˇˇˇ +2˙*1{2 +`p´3{2tEpR´4 +i qu1{4 +" +E +ˆˇˇˇ}ΓUi}3 ´ p3{2ˇˇˇ +2˙*1{2 +tPpAc +2iqu1{4 +À +ζ1p´1´δ{2 . +It follows that +Qjℓ “ n´1p´3{2 +nÿ +i“1 +ν´1 +i +pΓjUiqpΓℓUiq ` Oppζ1p´1´δ{2q . +For i “ 1, . . . , n, let A1i “ tp ´ ϵpp1`δq{2 ď }Ui}2 ď p ` ϵpp1`δq{2u for a fixed 0 ă ϵ ă 1. +According to Lemma A1, +E +”␣ +ΓjSpUiqSpUiqJΓJ +ℓ +(2ı +“ +E +␣ +}Ui}´4pΓjUiU J +i ΓJ +ℓ q2( +“ +E +␣ +}Ui}´4pΓjUiU J +i ΓJ +ℓ q2IpA1iq +( +` E +␣ +}Ui}´4pΓjUiU J +i ΓJ +ℓ q2IpAc +1iq +( +À +tp ´ ϵpp1`δq{2u´2E +␣ +pΓjUiU J +i ΓJ +ℓ q2( +` p2PpAc +1iq +49 + +À +tp ´ ϵpp1`δq{2u´2 ` p2 ˆ c1 expt´c2pδα{p4α`4qu +À +p´2 . +Then, we can show that +n´1p´3{2 +nÿ +i“1 +ν´1 +i +pΓjUiqpΓℓUiq +“ +n´1p´1{2 +nÿ +i“1 +ν´1 +i +ΓjSpUiqSpUiqJΓJ +ℓ ` Oppζ1p´7{6q , +where the last equality is indicated by +E|p´3{2ν´1 +i +ΓjSpUiqSpUiqJΓJ +ℓ p}Ui}2 ´ pq| +À +p´3{2tEpν´3 +i +qu1{3 ´ +E +”␣ +ΓjSpUiqSpUiqJΓJ +ℓ +(2ı¯1{2 “ +E +␣ +p}Ui}2 ´ pq6(‰1{6 +À +ζ1p´7{6 . +Thus, we obtain that +Qjℓ “ n´1p´1{2 +nÿ +i“1 +ν´1 +i +ΓjSpUiqSpUiqJΓJ +ℓ ` Oppζ1p´7{6 ` ζ1p´1´δ{2q . +As νi and SpUiq are independent with each other, we have +E +# +n´1p´1{2 +nÿ +i“1 +ν´1 +i +ΓjSpUiqSpUiqJΓJ +ℓ ++ +“ +p´1{2Epν´1 +i +qE +␣ +ΓjSpUiqSpUiqJΓJ +ℓ +( +, +where Epν´1 +i +q À p1{2ζ1 from Lemma A2. +According to Lemma A1 and regarding that ΓjΓJ +ℓ “ ωjℓ, +E +␣ +ΓjSpUiqSpUiqJΓJ +ℓ +( +“ +E +` +ΓjUiU J +i ΓJ +ℓ }Ui}´2˘ +“ +p´1E +` +ΓjUiU J +i ΓJ +ℓ +˘ +` E +␣ +ΓjUiU J +i ΓJ +ℓ +` +}Ui}´2 ´ p´1˘( +“ +p´1ωjℓ ` E +␣ +ΓjUiU J +i ΓJ +ℓ +` +}Ui}´2 ´ p´1˘( +50 + +ď +p´1|ωjℓ| ` E +`ˇˇΓjUiU J +i ΓJ +ℓ +ˇˇ ˇˇ}Ui}´2 ´ p´1ˇˇ˘ +“ +p´1|ωjℓ| ` p´1E +`ˇˇΓjUiU J +i ΓJ +ℓ +ˇˇ }Ui}´2 ˇˇ}Ui}2 ´ p +ˇˇ˘ +“ +p´1|ωjℓ| ` p´1E +␣ˇˇΓjUiU J +i ΓJ +ℓ +ˇˇ }Ui}´2 ˇˇ}Ui}2 ´ p +ˇˇ IpA1iq +( +`p´1E +␣ˇˇΓjUiU J +i ΓJ +ℓ +ˇˇ }Ui}´2 ˇˇ}Ui}2 ´ p +ˇˇ IpAc +1iq +( +À +p´1|ωjℓ| ` tp2 ´ ϵpp3`δq{2u´1E +`ˇˇΓjUiU J +i ΓJ +ℓ +ˇˇ ˇˇ}Ui}2 ´ p +ˇˇ˘ +`E +␣ˇˇ}Ui}2 ´ p +ˇˇ IpAc +1iq +( +ď +p´1|ωjℓ| ` tp2 ´ ϵpp3`δq{2u´1 ” +E +!` +ΓjUiU J +i ΓJ +ℓ +˘2)ı1{2 ” +E +!` +}Ui}2 ´ p +˘2)ı1{2 +` +” +E +!` +}Ui}2 ´ p +˘2)ı1{2 +tPpAc +1iqu1{2 +ď +p´1|ωjℓ| ` Opp´3{2q ` Opp1{2q ˆ c1{2 +1 +expt´c2pδα{p4α`4q{2u +À +p´1|ωjℓ| ` Opp´3{2q , +where the second last inequality is due to +E +!` +}Ui}2 ´ p +˘2) +“ +Ep}Ui}4 ´ 2p}Ui}2 ` p2q +“ +pEpU 4 +i,jq ` ppp ´ 1q ´ 2p2 ` p2 +“ +Oppq. +Thus, it follows that +E +# +n´1p´1{2 +nÿ +i“1 +ν´1 +i +ΓjSpUiqSpUiqJΓJ +ℓ ++ +À ζ1p´1|ωjj| ` Opζ1p´3{2q . +Furthermore, as Epν´2 +i +q À pζ2, we can conclude that +Var +# +n´1p´1{2 +nÿ +i“1 +ν´1 +i +ΓjSpUiqSpUiqJΓJ +ℓ ++ +“ +n´1p´1Epν´2 +i +qE +”␣ +ΓjSpUiqSpUiqJΓJ +ℓ +(2ı +´n´1p´1tEpν´1 +i +qu2 “ +E +␣ +ΓjSpUiqSpUiqJΓJ +ℓ +(‰2 +À +ζ2 +1n´1p´2. +51 + +It follows from the Chebychev’s inequality that +ˇˇˇˇˇn´1p´1{2 +nÿ +i“1 +ν´1 +i +ΓjSpUiqSpUiqJΓJ +ℓ +ˇˇˇˇˇ À ζ1p´1|ωjℓ| ` Oppζ1n´1{2p´1 ` ζ1p´3{2q . +Finally, we arrive at |Qj,ℓ| À ζ1p´1|ωjℓ| ` Oppζ1n´1{2p´1 ` ζ1p´7{6 ` ζ1p´1´δ{2q. +(ii) From the proof of part (i), we know that Qjℓ “ Q0,jℓ ` Oppζ1p´7{6 ` ζ1p´1´δ{2q , where +Q0,jℓ is the pj, ℓqth component of the random matrix Q0 “ n´1p´1{2 řn +i“1 ν´1 +i +tΓSpUiqutΓSpUiquJ. +In addition, E +␣ +ΓjSpUiqSpUiqJΓJ +ℓ +( +À p´1|ωjℓ| ` Opp´3{2q. It follows that +tr +!` +E +“ +tΓSpUiqutΓSpUiquJ‰˘2) +“ +pÿ +j“1 +pÿ +ℓ“1 +“ +E +␣ +ΓjSpUiqSpUiqJΓJ +ℓ +(‰2 +À +p´2 +pÿ +j“1 +pÿ +ℓ“1 +|ωjℓ|2 ` p´5{2 +pÿ +j“1 +pÿ +ℓ“1 +|ωjℓ| ` p´1 +À +p´1a0ppq ` p´3{2a0ppq ` p´1 +À +p´δ . +This implies that +trrtEpQ0qu2s +“ +p´1tEpν´1 +i +qu2tr +!` +E +“ +tΓSpUiqutΓSpUiquJ‰˘2) +À +p´1´δ +and +EttrpQ2 +0qu +“ +n´1p´1tr +` +E +“ +ν´2 +i +tΓSpUiqutΓSpUiquJtΓSpUiqutΓSpUiquJ‰˘ +`p1 ´ n´1qp´1tr +!` +E +“ +ν´1 +i +tΓSpUiqutΓSpUiquJ‰˘2) +“ +n´1p´1Epν´2 +i +qE +␣ +}ΓSpUiq}4( +`p1 ´ n´1qp´1tEpν´1 +i +qu2tr +!` +E +“ +tΓSpUiqutΓSpUiquJ‰˘2) +“ +Opn´1p´1q ` trrtEpQ0qu2sp1 ´ n´1q . +52 + +Thus, we have +trrEpQ2 +0q ´ tEpQ0qu2s “ Opn´1p´1q . +We complete the proof of this lemma. +Proof of Lemma A4. Recall that Γj is the jth row of Γ, and denote Γjℓ to be the pj, ℓqth element +of Γ, then +ΓjUi “ +pÿ +ℓ“1 +ΓjℓUi,ℓ. +It is noticed that ωjℓ “ řp +j1“1 Γjj1Γℓj1, then +VarpΓjUiq “ +pÿ +ℓ“1 +Γ2 +jℓ “ ωjj +and +EtpΓjUiq4u +“ +E +$ +& +% +˜ pÿ +ℓ“1 +ΓjℓUi,ℓ +¸4, +. +- +“ +pÿ +ℓ“1 +Γ4 +jℓEpU 4 +i,ℓq ` 6 +ÿ +1ďℓ1‰ℓ2ďp +Γ2 +jℓ1Γ2 +jℓ2EpU 2 +i,ℓ1qEpU 2 +i,ℓ2q +À +ω2 +jj . +(i) For i “ 1, . . . , n, let A2i “ tp1 ´ ϵqtrpΩq ď }ΓUi}2 ď p1 ` ϵqtrpΩqu for a fixed 0 ă ϵ ă 1, +then +PpA2iq ě 1 ´ c1 expt´c2pδα{p4α`4qu +according to the proof of Lemma A1. It follows that +EpW 4 +i,jq +“ +Et}ΓUi}´4pΓjUiq4u +“ +Et}ΓUi}´4pΓjUiq4IpA2iqu ` Et}ΓUi}´4pΓjUiq4IpAc +2iqu +53 + +ď +tp1 ´ ϵqtrpΩqu´2EtpΓjUiq4u ` PpAc +2iq +À +ω2 +jjtp1 ´ ϵqtrpΩqu´2 ` c1 expt´c2pδα{p4α`4qu +À +ω2 +jjttrpΩqu´2 +and +EpW 2 +i,jq +ě +Et}ΓUi}´2pΓjUiq2IpA2iqu +ě +tp1 ` ϵqtrpΩqu´1EtpΓjUiq2IpA2iqu +“ +tp1 ` ϵqtrpΩqu´1EtpΓjUiq2u ´ tp1 ` ϵqtrpΩqu´1EtpΓjUiq2IpAc +2iqu +ě +tp1 ` ϵqtrpΩqu´1EtpΓjUiq2u ´ tp1 ` ϵqtrpΩqu´1rEtpΓjUiq4us1{2tPpAc +2iqu1{2 +Á +ωjjtp1 ` ϵqtrpΩqu´1 ´ tp1 ` ϵqtrpΩqu´1 ˆ ωjj ˆ c1{2 +1 +expt´c2pδα{p4α`4q{2u +Á +ωjjttrpΩqu´1, +from which we conclude that +Etpζ´1 +1 Wi,jq4u À ζ´4 +1 p´2ω2 +jj À ¯ +M2 +and +Etpζ´1 +1 Wi,jq2u Á ζ´2 +1 p´1ωjj Á m. +(ii) Similar to the proof of part (i), for any ϱ ě 1, +E +␣ +|ζ´1 +1 Wi,j|ϱ( +“ +E +␣ +|ζ´1 +1 Wi,j|ϱIpA1iq +( +` E +␣ +|ζ´1 +1 Wi,j|ϱIpAc +1iq +( +À +ζ´ϱ +1 ttrpΩqu´ϱ{2Et|ΓjUi|ϱu ` ζ´ϱ +1 PpAc +1iq +À +Et|ΓjUi|ϱu ` pϱ{2 expt´c2pδα{p4α`4qu. +Since max1ďjďp }Ui,j}ψα ď c0 for some constant c0, we have }ΓjUi}ψα À c0 according to Lemma +B.4 in Koike (2021). Then, we known that Et|ΓjUi|ϱu À ϱϱ{α for any ϱ ě 1 by the equivalent +54 + +sub-exponential properties (Koike 2021). Therefore, +E +␣ +|ζ´1 +1 Wi,j|ϱ( +À +ϱϱ{α +for any ϱ ě 1 for sufficient large p, which indicates that ζ´1 +1 Wi,j is sub-exponential, and thus +}ζ´1 +1 Wi,j}ψα À ¯B. +(iii) By simple algebra, +EpW 2 +i,jq +“ +p´1EtpΓjUiq2u ` EtpΓjUiq2p}ΓUi}´2 ´ p´1qu +“ +p´1ωjj ` EtpΓjUiq2p}ΓUi}´2 ´ p´1qu , +where EtpΓjUiq2p}ΓUi}´2 ´ p´1qu satisfies +ˇˇEtpΓjUiq2p}ΓUi}´2 ´ p´1qu +ˇˇ +ď +p´1EtpΓjUiq2}ΓUi}´2|}ΓjUi}2 ´ p|u +“ +p´1EtpΓjUiq2}ΓUi}´2|}ΓjUi}2 ´ p|IpA2iqu +`p´1EtpΓjUiq2}ΓUi}´2|}ΓjUi}2 ´ p|IpAc +2iqu +ď +p´1tp1 ´ ϵqtrpΩqu´1EtpΓjUiq2|}ΓjUi}2 ´ p|u ` p´1Et}ΓjUi}2 ´ p|IpAc +2iqu +ď +p´2p1 ´ ϵq´1rEtpΓjUiq4us1{2tEp|}ΓjUi}2 ´ p|2qu1{2 +`p´1tEp|}ΓjUi}2 ´ p|2qu1{2tPpAc +2iqu1{2 +À +p´2 ˆ p1´δ{2 ` p´1 ˆ p1´δ{2 ˆ c1{2 +1 +expt´c2pδα{p4α`4q{2u +À +p´1´δ{2. +In addition, for 1 ď j ‰ ℓ ď p, we have +EpWi,jWi,ℓq +“ +p´1EtpΓjUiqpΓℓUiqu ` EtpΓjUiqpΓℓUiqp}ΓUi}´2 ´ p´1qu +“ +p´1ωjℓ ` EtpΓjUiqpΓℓUiqp}ΓUi}´2 ´ p´1qu , +55 + +where EtpΓjUiqpΓℓUiqp}ΓUi}´2 ´ p´1qu satisfies +ˇˇEtpΓjUiqpΓℓUiqp}ΓUi}´2 ´ p´1qu +ˇˇ +ď +p´1Et|pΓjUiqpΓℓUiq|}ΓUi}´2|}ΓjUi}2 ´ p|u +“ +p´1Et|pΓjUiqpΓℓUiq|}ΓUi}´2|}ΓjUi}2 ´ p|IpA2iqu +`p´1Et|pΓjUiqpΓℓUiq|}ΓUi}´2|}ΓjUi}2 ´ p|IpAc +2iqu +ď +p´1tp1 ´ ϵqtrpΩqu´1Et|pΓjUiqpΓℓUiq||}ΓjUi}2 ´ p|u ` p´1Et}ΓjUi}2 ´ p|IpAc +2iqu +ď +p´2p1 ´ ϵq´1rEt|pΓjUiqpΓℓUiq|2us1{2tEp|}ΓjUi}2 ´ p|2qu1{2 +`p´1tEp|}ΓjUi}2 ´ p|2qu1{2tPpAc +2iqu1{2 +À +p´2 ˆ p1´δ{2 ` p´1 ˆ p1´δ{2 ˆ c1{2 +1 +expt´c2pδα{p4α`4q{2u +À +p´1´δ{2. +(iv) According to part (ii), ζ´1 +1 W1, . . . , ζ´1 +1 Wn are i.i.d. p-dimensional random vectors satis- +fies }ζ´1 +1 Wi,j}ψα À ¯B for all i “ 1, . . . , n and j “ 1, . . . , p. By Lemma 2.2.2 of van der Vaart & +Wellner (1996), +›››› max +1ďiďn max +1ďjďp |ζ´1 +1 Wi,j| +›››› +ψα +À log1{αpnpq . +Similar to the proof of part (i), we can show that +Etpζ´1 +1 Wi,jq2u +“ +ζ´2 +1 Et}ΓUi}´2pΓjUiq2IpA1iqu +`ζ´2 +1 Et}ΓUi}´4pΓjUiq4IpAc +1iqu +ď +ζ´2 +1 tp1 ` ϵqtrpΩqu´1EtpΓjUiq2u ` ζ´2 +1 EtIpAc +1iqu +ď +ζ´2 +1 ωjjtp1 ` ϵqtrpΩqu´1 ` ζ´2 +1 c1 expt´c2pδ{p4`4αqu +“ +ζ´2 +1 ωjjtp1 ` ϵqtrpΩqu´1t1 ` op1qu . +It follows that +max +1ďjďp +nÿ +i“1 +Etpζ´1 +1 Wi,jq2u +À +max +1ďjďp +nÿ +i“1 +ζ´2 +1 ωjjtp1 ` ϵqtrpΩqu´1 À n max +1ďjďp ωjj ď ¯ +Mn , +56 + +Applying Lemma E.1 of Chernozhukov, Chetverikov and Kato (2017), it holds that with α ě 1 +and n´1{2 log3{2pnpq À 1, +E +˜ˇˇˇˇˇn´1{2 +nÿ +i“1 +ζ´1 +1 Wi +ˇˇˇˇˇ +8 +¸ +À +n´1{2tn1{2 log1{2ppq ` log1{αpnpq logppqu +À +log1{2pnpq . +From the properties of the ψα norm, it holds that +›››› +max +1ďiďn,1ďjďp |ζ´1 +1 Wi,j|2 +›››› +ψα{2 +À log2pnpq. +According to Lemma E.3 of Chernozhukov, Chetverikov and Kato (2017), we have that +E +˜ˇˇˇˇˇn´1 +nÿ +i“1 +pζ´1 +1 Wiq2 +ˇˇˇˇˇ +8 +¸ +À +n´1t ¯ +Mn ` log2pnpq logppqu À ¯ +M . +We finish the proof of this lemma. +Proof of Lemma A5. Let ˜Xi “ Xi ´ ˆθn and ˜Ri “ } ˜Xi} for i “ 1, . . . , n. According to the proof +of Lemma 1, }ˆθn} “ Oppζ´1 +1 n´1{2q and max1ďiďn R´1 +i +“ Oppζ1n1{4q. Then R´1 +i }ˆθn} satisfies +R´1 +i }ˆθn} “ Oppn´1{2q and +max +1ďiďn R´1 +i }ˆθn} “ Oppn´1{4q . +As ˜R´1 +i +“ R´1 +i }Wi ´R´1 +i +ˆθn}´1 “ R´1 +i +´ +1 ´ 2R´1 +i W J +i ˆθn ` R´2 +i }ˆθn}2¯´1{2 +, by Taylor expansion, +˜R´1 +i +“ R´1 +i +´ +1 ` R´1 +i W J +i ˆθn ´ 2´1R´2 +i }ˆθn}2 ` ˜δ1i +¯ +, +where ˜δ1i satisfies ˜δ1i “ Oppn´1q and max1ďiďn ˜δ1i “ Oppn´1{2q. It follows that +˜R´1 +i +“ R´1 +i p1 ` ˜δ2iq , +where ˜δ2i “ R´1 +i W J +i ˆθn ´ 2´1R´2 +i }ˆθn}2 ` ˜δ1i satisfies ˜δ2i “ Oppn´1{2q and max1ďiďn ˜δ2i “ +57 + +Oppn´1{4q. Thus, +˜R´1 +i +“ Oppζ1q and +max +1ďiďn +˜R´1 +i +“ Oppζ1n1{4q . +Denote ˜Wi “ ˜Xi{} ˜Xi} for i “ 1, . . . , n. Then, +˜Wi +“ +˜R´1 +i pXi ´ ˆθnq +“ +R´1 +i pXi ´ ˆθnqp1 ` ˜δ2iq +“ +pWi ´ R´1 +i +ˆθnqp1 ` ˜δ2iq . +We first show that }˜θn} “ Oppζ´1 +1 n´1{2q. It is noticed that ˜θn minimizes +L˚ +npβq “ +nÿ +i“1 +}Zi ˜Xi ´ β} , +which is a strictly convex function of β. Thus, if we can show that L˚ +npβq has a ζ1n1{2-consistent +local minimizer, then this local minimizer must be a ζ1n1{2-consistent global minimizer of L˚ +npβq. +The existence of a ζ1n1{2-consistent local minimizer is implied by the fact that for an arbitrarily +small ε ą 0, there exists a constant C0, which does not depend on n and p, such that +lim inf +n +P +" +inf +qPRp, }q}“C0 +L˚ +npζ´1 +1 n´1{2qq ą L˚ +np0q +* +ą 1 ´ ε, +(S.20) +Since |Zi| “ 1, we rewrite }Zi ˜Xi ´ ζ´1 +1 n´1{2q} as +}Zi ˜Xi ´ ζ´1 +1 n´1{2q} +“ +˜Ri +´ +1 ´ 2ζ´1 +1 n´1{2 ˜R´1 +i ZiqJ ˜Wi ` ζ´2 +1 n´1 ˜R´2 +i }q}2¯1{2 +. +As |ζ´1 +1 n´1{2 ˜R´1 +i ZiqT ˜Wi| “ Oppn´1{2q and ζ´2 +1 n´1 ˜R´2 +1i }q}2 “ Oppn´1q, by Taylor expansion, +we obtain that +}Zi ˜Xi ´ ζ´1 +1 n´1{2q} +“ +˜Ri ´ ζ´1 +1 n´1{2ZiqJ ˜Wi ` 2´1ζ´2 +1 n´1 ˜R´1 +i }q}2 +58 + +´2´1ζ´2 +1 n´1 ˜R´1 +i qJ ˜Wi ˜W J +i q ` Oppζ´1 +1 n´3{2q . +Then, +ζ1 +! +L˚ +npζ´1 +1 n´1{2qq ´ L˚ +np0q +) +“ +ζ1 +nÿ +i“1 +´ +}Zi ˜Xi ´ ζ´1 +1 n´1{2q} ´ } ˜Xi} +¯ +“ +´n´1{2qJ +˜ nÿ +i“1 +Zi ˜Wi +¸ +` 2´1ζ´1 +1 n´1}q}2 +nÿ +i“1 +˜R´1 +i +´2´1ζ´1 +1 n´1qJ +˜ nÿ +i“1 +˜Ri ˜Wi ˜W J +i +¸ +q ` Oppn´1{2q . +(S.21) +As E˚ ´ +n´1{2 řn +i“1 Zi ˜Wi +¯ +“ 0 and +E˚ +¨ +˝ +›››››n´1{2 +nÿ +i“1 +Zi ˜Wi +››››› +2˛ +‚“ n´1 +nÿ +i“1 +˜W J +i ˜Wi “ 1, +we obtain that +ˇˇˇˇˇn´1{2qJ +nÿ +i“1 +Zi ˜Wi +ˇˇˇˇˇ ď }q} +›››››n´1{2 +nÿ +i“1 +Zi ˜Wi +››››› “ Opp}q}q . +In the meanwhile, as ζ´1 +1 n´1 řn +i“1 R´1 +i +“ 1 ` Oppn´1{2q, we have +ζ´1 +1 n´1}q}2 +nÿ +i“1 +˜R´1 +i +“ +ζ´1 +1 n´1}q}2 +nÿ +i“1 +R´1 +i p1 ` ˜δ2iq +“ +}q}2t1 ` Oppn´1{4qu . +Simple algebra yields +n´1 +nÿ +i“1 +˜R´1 +i +˜Wi ˜W J +i +“ +n´1 +nÿ +i“1 +R´1 +i pWi ´ R´1 +i +ˆθnqpWi ´ R´1 +i +ˆθnqJp1 ` ˜δ2iq +59 + +“ +n´1 +nÿ +i“1 +RiWiW J +i p1 ` ˜δ2iq ´ 2n´1 +nÿ +i“1 +R´2 +i Wiˆθ +J +np1 ` ˜δ2iq +`n´1 ÿ +i“1 +R´3 +i +ˆθnˆθ +J +np1 ` ˜δ2iq . +Similar to the proof in Cheng et.al (2019) and utilizing the results on Q “ n´1 řn +i“1 R´1 +i WiW ´1 +i +in Lemma A3, we can show that n´1qJ řn +i“1 RiWiW J +i qp1 ` ˜δ2iq “ Oppζ1n´1{2 ` ζ1p´p1{6^δ{2qq. +In addition, as +n´1 +nÿ +i“1 +R´2 +i qJWi ď n´1 +nÿ +i“1 +R´2 +i }q}}Wi} “ }q}n´1 +nÿ +i“1 +R´2 +i +“ Oppζ2 +1q +and n´1 řn +i“1 R´3 +i +“ ζ3t1 ` opp1qu, we have +n´1qJ +nÿ +i“1 +R´2 +i Wiˆθ +J +nqp1 ` ˜δ2iq +“ +n´1 +nÿ +i“1 +R´2 +i qJWip1 ` ˜δ2iqpˆθ +J +nqq “ Oppζ1n´1{2q . +and +n´1qJ ÿ +i“1 +R´3 +i +ˆθnˆθ +J +nqp1 ` ˜δ2iq “ n´1 ÿ +i“1 +R´3 +i p1 ` ˜δ2iq}qJˆθn}2 “ Oppζ1n´1q . +Thus, we obtain +2´1ζ´1 +1 n´1}q}2 +nÿ +i“1 +˜R´1 +i +` 2´1ζ´1 +1 n´1qJ +˜ nÿ +i“1 +˜Ri ˜Wi ˜W J +i +¸ +q +“ +2´1}q}2 ` Oppn´1{4 ` p´δq . +Choosing a sufficient large constant C0, the second term dominates the first term in (S.21) and +thus ζ1 +␣ +L˚ +npζ´1 +1 n´1{2qq ´ L˚ +np0q +( +ą 0. Hence, we have }˜θn} “ Oppζ´1 +1 n´1{2q. +Denote Θi “ Ziˆθn ` ˜θn for i “ 1, . . . , n. Then +max +1ďiďn }Θi} ď }ˆθn} ` }˜θn} “ Oppζ´1 +1 n´1{2q. +60 + +Recall that ˜θn satisfies +nÿ +i“1 +Zi ˜Xi ´ ˜θn +}Zi ˜Xi ´ ˜θn} +“ +nÿ +i“1 +ZiWi ´ R´1 +i Θi +}ZiWi ´ R´1 +i Θi} “ 0 , +which is equivalently to +n´1 +nÿ +i“1 +pZiWi ´ R´1 +i Θiq +` +1 ´ 2ZiR´1 +i W J +i Θi ` R´2 +i }Θi}2˘´1{2 “ 0 , +where |R´1 +i W J +i Θi| “ Oppn´1{2q, R´2 +i }Θi}2 “ Oppn´1q, +max +1ďiďn |R´1 +i W J +i Θi| “ Oppn´1{4q and +max +1ďiďn R´2 +i }Θi}2 “ Oppn´1{2q . +Taylor expansion leads to +n´1 +nÿ +i“1 +pZiWi ´ R´1 +i Θiqp1 ` ZiR´1 +i W J +i Θi ´ 2R´2 +i }Θi}2 ` ˜δ3iq “ 0 +where δ3i “ OptpZiR´1 +i W J +i Θi ´ R´2 +i }Θi}2q2u “ Oppn´1q, and max1ďiďn δ3i “ Oppn´1{2q. Then, +n´1 +nÿ +i“1 +ZiWip1 ´ 2R´2 +i }Θi}2 ` ˜δ3iq ` n´1 +nÿ +i“1 +R´1 +i pW J +i ΘiqWi +“ +n´1 +nÿ +i“1 +ZiWip1 ´ 2R´2 +i }Θi}2 ` ˜δ3iq ` n´1 +nÿ +i“1 +ZiR´1 +i WiW J +i ˆθn +`n´1 +nÿ +i“1 +R´1 +i WiW J +i ˜θn +“ +n´1 +nÿ +i“1 +R´1 +i Θip1 ` ˜δ3i ` ˜δ4iq +“ +n´1 +nÿ +i“1 +R´1 +i +˜θnp1 ` ˜δ3i ` ˜δ4iq ` n´1 +nÿ +i“1 +ZiR´1 +i +ˆθnp1 ` ˜δ3i ` ˜δ4iq , +where ˜δ4i “ ZiR´1 +i W J +i Θi ´ 2R´2 +i }Θi}2 “ Opp˜δ1{2 +3i q satisfies max1ďiďn ˜δ4i “ Oppn´1{4q. +The proof of Lemma 1 implies |ˆθ|8 “ Optn´1{2 log1{2pnpqu. As E˚ ` +n´1 řn +i“1 ZiR´1 +i +˘ +“ +0 and E˚ !` +n´1 řn +i“1 ZiR´1 +i +˘2) +“ n´2 řn +i“1 R´2 +i +“ Oppn´1ζ2q, we have n´1 řn +i“1 ZiR´1 +i +“ +Oppζ1n´1{2q. +61 + +As Zi is bounded, it is straightforward to show that |n´1{2 řn +i“1 ZiWi|8 “ Optp´1{2 log1{2pnpqu +similar as in the proof of Lemma A4 (iii). Thus, similar to the proof of Lemma 1, we obtain +that +|˜θ|8 “ Optn´1{2 log1{2pnpqu +and +ˇˇˇˇˇn´1 +nÿ +i“1 +R´1 +i WiW J +i ˜θn +ˇˇˇˇˇ +8 +“ +Optζ1n´1{2p´p1{6^δ{2q log1{2pnpq ` ζ1n´1 log1{2pnpqu . +In the meanwhile, it holds that |n´1 řn +i“1 R´1 +i | “ ζ1 ` Oppζ1n´1{2q. Finally, +n1{2˜θn “ n´1{2ζ´1 +1 +nÿ +i“1 +ZiWi ` ˜Cn , +(S.22) +and ˜Cn is the remainder term satisfies +| ˜Cn|8 “ Optn´1{4 log1{2pnpq ` p´δ´p1{6^δ{2q log1{2pnpqu . +We finish the proof of this lemma. +Appendix C: Additional simulation results +In this section, we report additional simulation results. Section C1 presents simulation results +on SCIs for ρ “ 0.2 and 0.5. Section C2 reports simulations on global tests for high-dimensional +location parameters. +C.1 +Addition simulation results on simultaneous confidence intervals +Tables A4 reports the coverage probability and median length of the SCIs based on ˆθn for ρ “ 0.2 +and 0.5, the results of the SCIs based on the sample mean ¯Xn are presented in parentheses. +We observe that the performance of the SCIs based on ˆθn with ρ “ 0.2 and 0.5 is similar to +62 + +that of ρ “ 0.0 and 0.8 in the main paper. The SCIs achieve satisfactory coverage probability, +and it is much shorter than those based on ¯Xn under the multivariate t-distribution, which is +heavy-tailed. +Table A4: Coverage probability (in %) and median length of the SCIs based on ˆθn, the results +of the SCIs based on ¯Xn are in parentheses. +θ “ θ1 +θ “ θ2 +Coverage probability +Median length +Coverage probability +Median length +Model +ρ +n +p +90% +95% +90% +95% +90% +95% +90% +95% +I +0.2 100 +100 89.8 (89.9) 94.5 (94.5) 0.65 (0.65) 0.69 (0.69) +88.8 (88.7) 94.4 (94.4) 0.65 (0.65) 0.69 (0.69) +1000 88.7 (88.7) 94.5 (94.3) 0.77 (0.77) 0.80 (0.80) +90.0 (89.6) 94.7 (94.8) 0.77 (0.77) 0.80 (0.80) +200 +100 89.0 (88.9) 94.3 (94.1) 0.46 (0.46) 0.49 (0.49) +88.8 (88.8) 94.0 (94.2) 0.46 (0.46) 0.49 (0.49) +1000 89.8 (89.8) 94.4 (94.4) 0.55 (0.55) 0.57 (0.57) +88.7 (89.2) 94.6 (94.3) 0.55 (0.55) 0.57 (0.57) +0.5 100 +100 89.6 (89.8) 94.5 (94.4) 0.65 (0.65) 0.69 (0.69) +88.4 (88.8) 94.0 (94.1) 0.65 (0.65) 0.69 (0.69) +1000 88.4 (88.4) 94.3 (94.3) 0.77 (0.77) 0.80 (0.80) +87.4 (87.4) 94.1 (94.2) 0.77 (0.77) 0.80 (0.80) +200 +100 90.9 (90.9) 95.1 (95.2) 0.46 (0.46) 0.49 (0.49) +89.7 (90.0) 95.3 (95.0) 0.46 (0.46) 0.49 (0.49) +1000 89.0 (89.0) 94.2 (94.3) 0.55 (0.55) 0.57 (0.57) +88.8 (88.6) 94.3 (94.0) 0.55 (0.55) 0.57 (0.57) +II +0.2 100 +100 89.2 (88.8) 94.8 (94.2) 0.71 (1.05) 0.75 (1.12) +88.4 (88.8) 93.7 (94.3) 0.71 (1.05) 0.75 (1.11) +1000 89.0 (89.4) 94.1 (94.8) 0.84 (1.24) 0.88 (1.30) +89.0 (88.9) 94.4 (94.6) 0.84 (1.24) 0.88 (1.30) +200 +100 90.7 (89.8) 95.3 (94.7) 0.50 (0.76) 0.53 (0.80) +89.2 (89.7) 94.0 (94.4) 0.50 (0.76) 0.53 (0.80) +1000 88.6 (89.5) 94.2 (94.6) 0.59 (0.90) 0.62 (0.93) +89.0 (90.6) 95.0 (95.1) 0.59 (0.90) 0.62 (0.94) +0.5 100 +100 89.2 (87.9) 93.6 (93.9) 0.71 (1.05) 0.75 (1.12) +89.4 (88.6) 94.6 (94.1) 0.71 (1.05) 0.75 (1.11) +1000 89.2 (88.9) 94.4 (94.2) 0.84 (1.24) 0.88 (1.30) +90.0 (89.4) 94.7 (94.6) 0.84 (1.25) 0.88 (1.30) +200 +100 89.4 (90.0) 94.1 (94.6) 0.50 (0.76) 0.53 (0.80) +89.7 (88.6) 95.0 (93.6) 0.50 (0.76) 0.53 (0.80) +1000 90.0 (89.9) 95.6 (94.8) 0.59 (0.90) 0.62 (0.94) +88.8 (89.5) 93.8 (94.4) 0.59 (0.89) 0.62 (0.93) +III +0.2 100 +100 89.6 (89.5) 95.0 (95.1) 0.65 (0.66) 0.69 (0.70) +89.4 (89.4) 94.6 (94.6) 0.65 (0.66) 0.69 (0.70) +1000 89.3 (88.8) 94.5 (94.5) 0.78 (0.78) 0.82 (0.82) +90.3 (90.7) 95.0 (94.9) 0.78 (0.78) 0.82 (0.82) +200 +100 89.2 (89.0) 94.4 (94.4) 0.46 (0.46) 0.49 (0.49) +90.0 (89.6) 95.1 (95.2) 0.46 (0.46) 0.49 (0.49) +1000 89.7 (89.7) 94.6 (94.8) 0.55 (0.55) 0.57 (0.58) +90.4 (90.6) 95.0 (95.0) 0.55 (0.55) 0.57 (0.57) +0.5 100 +100 88.9 (89.3) 94.0 (94.6) 0.65 (0.65) 0.69 (0.69) +88.0 (88.5) 94.2 (94.0) 0.65 (0.65) 0.69 (0.69) +1000 89.1 (89.2) 94.3 (94.2) 0.78 (0.78) 0.81 (0.81) +89.2 (88.9) 94.1 (94.0) 0.78 (0.78) 0.81 (0.81) +200 +100 89.6 (89.7) 95.0 (94.4) 0.46 (0.46) 0.49 (0.49) +89.6 (89.7) 94.9 (94.4) 0.46 (0.46) 0.49 (0.49) +1000 89.0 (89.1) 94.3 (94.4) 0.55 (0.55) 0.57 (0.57) +89.3 (89.6) 95.4 (95.0) 0.55 (0.55) 0.57 (0.57) +C.2 +Simulations on global tests for high-dimensional location parameters +In this section, we report the performance of the test based on Tn (Median test) for one-sample +high-dimensional location parameters, and compare it with three alternative approaches: the +test of Chen and Qin +(2010, CQ test); the test based on TMean (Mean test) and bootstrap +approximation for ¯Xn; the test of Wang, Peng and Li (2015, WPL test) based on TWPL. We +consider the same data generation models (I, II and III) as in Section 5.1. For θ, we set its first +tc0 log pu components as non-zero, while the other elements are all zero. c0 is chosen from 0.5 +and 1. The magnitude of non-zero entries in θ is κplog p{nq1{2, where κ is chosen from 0 to 5. +Note that κ “ 0 refers to the null hypothesis. We consider n “ 50 or 100, and p “ 100 and 1000 +for each sample size. +63 + +Figures A3–A10 plot the empirical size (κ “ 0) and power (κ ‰ 0) of four (CQ, Mean, +Median, and WPL) tests at the 5% significance level for Models I and II. The results of κ “ 0 +indicates that the empirical sizes of all these four tests are close to the nominal significance level +under different case scenarios. When κ ‰ 0, the power of these tests increases as κ increases, +that is, as the signal getting stronger. For Gaussian data, the Mean test based on TMean and +the Median test based on Tn have similar power performances, and they advance both the +CQ test and the WPL test, which are L2-norm type tests. In addition, when the data are +from multivariate t-distribution, the Median test outperforms the Mean test, which shows the +superiority of the procedure based on the sample spatial median over that based on the sample +mean under heavy-tailedness. In summary, the Median test based on Tn is preferred among the +four tests when the alternative is sparse and the underlying distribution is heavy-tailed. +Second, Figure A11 depicts empirical size and power of the four tests (CQ, Mean, Median, +WPL) for Model III with ρ “ 0. It can be seen that, even Model III is not a member of the +elliptical distribution family, the size of the Median test can still control the size at the nominal +level α “ 0.05, and this is also the case for the WPL test. We can also see that the Median +test and the Mean test have better power performance than the CQ test and the WPL test, +especially for c0 “ 0.5 when the number of non-zero element in θ is relatively small. +References +Belloni, A., Chernozhukov, V., Chetverikov, D., Hansen, C., and Kato, K. (2018) High- +dimensional econometrics and generalized GMM. arXiv preprint arXiv:1806.01888. +Chen, S. X. and Qin, Y. (2010) A two-sample test for high-dimensional data with applications +to gene-set testing, Ann. Statist. 38 (2), 808–835. +Cheng,G. Liu, B. Peng, L; Zhang, B and Zheng, S. (2019).Testing the equality of two high- +dimensional spatial sign covariance matrices. Scand J Statist. 46, 257–271. +Chernozhukov, V., Chetverikov, D., and Kato, K. (2017). Central limit theorems and bootstrap +in high dimensions. The Annals of Probability. 45(4), 2309–2352. +64 + +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 50 , p = 100 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 50 , p = 1000 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 100 , p = 100 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 100 , p = 1000 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 50 , p = 100 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 50 , p = 1000 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 100 , p = 100 +0.00 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 100 , p = 1000 +Method +CQ +Mean +Median +WPL +Figure A3: Empirical size and power of the four tests (CQ, Mean, Median, WPL) for Models +I and II with c0 “ 0.5 and ρ “ 0. The horizontal black solid line refers to the nominal 5% +significance level. “Gaussian” denotes the multivariate normal distribution, and t3 denotes the +multivariate t-distribution with 3 degrees of freedom. +65 + +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 50 , p = 100 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 50 , p = 1000 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 100 , p = 100 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 100 , p = 1000 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 50 , p = 100 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 50 , p = 1000 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 100 , p = 100 +0.00 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 100 , p = 1000 +Method +CQ +Mean +Median +WPL +Figure A4: Empirical size and power of the four tests (CQ, Mean, Median, WPL) for Models +I and II with c0 “ 0.5 and ρ “ 0.2. The horizontal black solid line refers to the nominal 5% +significance level. “Gaussian” denotes the multivariate normal distribution, and t3 denotes the +multivariate t-distribution with 3 degrees of freedom. +66 + +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 50 , p = 100 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 50 , p = 1000 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 100 , p = 100 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 100 , p = 1000 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 50 , p = 100 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 50 , p = 1000 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 100 , p = 100 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 100 , p = 1000 +Method +CQ +Mean +Median +WPL +Figure A5: Empirical size and power of the four tests (CQ, Mean, Median, WPL) for Models +I and II with c0 “ 0.5 and ρ “ 0.5. The horizontal black solid line refers to the nominal 5% +significance level. “Gaussian” denotes the multivariate normal distribution, and t3 denotes the +multivariate t-distribution with 3 degrees of freedom. +67 + +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 50 , p = 100 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 50 , p = 1000 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 100 , p = 100 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 100 , p = 1000 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 50 , p = 100 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 50 , p = 1000 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 100 , p = 100 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 100 , p = 1000 +Method +CQ +Mean +Median +WPL +Figure A6: Empirical size and power of the four tests (CQ, Mean, Median, WPL) for Models +I and II with c0 “ 0.5 and ρ “ 0.8. The horizontal black solid line refers to the nominal 5% +significance level. “Gaussian” denotes the multivariate normal distribution, and t3 denotes the +multivariate t-distribution with 3 degrees of freedom. +68 + +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 50 , p = 100 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 50 , p = 1000 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 100 , p = 100 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 100 , p = 1000 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 50 , p = 100 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 50 , p = 1000 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 100 , p = 100 +0.00 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 100 , p = 1000 +Method +CQ +Mean +Median +WPL +Figure A7: Empirical size and power of the four tests (CQ, Mean, Median, WPL) for Models I +and II with c0 “ 1 and ρ “ 0. The horizontal black line refers to the nominal 5% significance +level. “Gaussian” denotes the multivariate normal distribution, and t3 denotes the multivariate +t-distribution with 3 degrees of freedom. +69 + +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 50 , p = 100 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 50 , p = 1000 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 100 , p = 100 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 100 , p = 1000 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 50 , p = 100 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 50 , p = 1000 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 100 , p = 100 +0.00 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 100 , p = 1000 +Method +CQ +Mean +Median +WPL +Figure A8: Empirical size and power of the four tests (CQ, Mean, Median, WPL) for Models +I and II with c0 “ 1 and ρ “ 0.2. The horizontal black solid line refers to the nominal 5% +significance level. “Gaussian” denotes the multivariate normal distribution, and t3 denotes the +multivariate t-distribution with 3 degrees of freedom. +70 + +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 50 , p = 100 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 50 , p = 1000 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 100 , p = 100 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 100 , p = 1000 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 50 , p = 100 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 50 , p = 1000 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 100 , p = 100 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 100 , p = 1000 +Method +CQ +Mean +Median +WPL +Figure A9: Empirical size and power of the four tests (CQ, Mean, Median, WPL) for Models +I and II with c0 “ 1 and ρ “ 0.5. The horizontal black solid line refers to the nominal 5% +significance level. “Gaussian” denotes the multivariate normal distribution, and t3 denotes the +multivariate t-distribution with 3 degrees of freedom. +71 + +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 50 , p = 100 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 50 , p = 1000 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 100 , p = 100 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +Gaussian, n = 100 , p = 1000 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 50 , p = 100 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 50 , p = 1000 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 100 , p = 100 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +t3 , n = 100 , p = 1000 +Method +CQ +Mean +Median +WPL +Figure A10: Empirical size and power of the four tests (CQ, Mean, Median, WPL) for Models I +and II with c0 “ 1 and ρ “ 0.8. The horizontal black line refers to the nominal 5% significance +level. “Gaussian” denotes the multivariate normal distribution, and t3 denotes the multivariate +t-distribution with 3 degrees of freedom. +72 + +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +n = 50 , p = 100 , c0 = 0.5 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +n = 50 , p = 100 , c0 = 1 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +n = 50 , p = 1000 , c0 = 0.5 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +n = 50 , p = 1000 , c0 = 1 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +n = 100 , p = 100 , c0 = 0.5 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +n = 100 , p = 100 , c0 = 1 +0.00 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +n = 100 , p = 1000 , c0 = 0.5 +0.00 +0.25 +0.50 +0.75 +1.00 +0 +1 +2 +3 +4 +5 +κ +Rejection Probability +n = 100 , p = 1000 , c0 = 1 +Method +CQ +Mean +Median +WPL +Figure A11: Empirical size and power of the four tests (CQ, Mean, Median, WPL) for Model +III with ρ “ 0. The horizontal black solid line refers to the nominal 5% significance level. +73 + +Koike, Y. (2021). Notes on the dimension dependence in high-dimensional central limit theorems +for hyperrectangles. Japanese Journal of Statistics and Data Science. 1, 257–297. +Liu, W. and Shao, Q.-M. (2014). Phase transition and regularized bootstrap in large scale t-tests +with false discovery rate control. Annals of Statistics. 42, 2003–2025. +Rudelson, M., and Vershynin, R. (2013) Hanson–Wright inequality and sub-Gaussian concen- +tration. Electronic Communications in Probability. 18, 1–9. +Vershynin, R. (2018). High-Dimensional Probability. Cambridge University Press, 2018. +Wang, L., Peng, B. and Li, R. (2015). A high-dimensional nonparametric multivariate test for +mean vector. Journal of the American Statistical Association. 110, 1658–1669. +74 + diff --git a/G9E1T4oBgHgl3EQfXQT3/content/tmp_files/load_file.txt b/G9E1T4oBgHgl3EQfXQT3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c94014aab32fc8d194ac9073d8466c113d9852da --- /dev/null +++ b/G9E1T4oBgHgl3EQfXQT3/content/tmp_files/load_file.txt @@ -0,0 +1,2800 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf,len=2799 +page_content='Statistical Inference for Ultrahigh Dimensional Location Parameter Based on Spatial Median Guanghui Chenga,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Liuhua Pengb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Changliang Zouc aGuangzhou Institute of International Finance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Guangzhou University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' bSchool of Mathematics and Statistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The University of Melbourne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' c School of Statistics and Data Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Nankai University Abstract Motivated by the widely used geometric median-of-means estimator in machine learning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' this paper studies statistical inference for ultrahigh dimensionality location parameter based on the sample spatial median under a general multivariate model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' including simultaneous confidence intervals construction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' global tests,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' and multiple testing with false discovery rate control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' To achieve these goals, we derive a novel Bahadur representation of the sample spa- tial median with a maximum-norm bound on the remainder term, and establish Gaussian approximation for the sample spatial median over the class of hyperrectangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In addi- tion, a multiplier bootstrap algorithm is proposed to approximate the distribution of the sample spatial median.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The approximations are valid when the dimension diverges at an exponentially rate of the sample size, which facilitates the application of the spatial median in the ultrahigh dimensional region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The proposed approaches are further illustrated by simulations and analysis of a genomic dataset from a microarray study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' keywords: Bootstrap approximation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Gaussian approximation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' high-dimensional;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' spatial me- dian;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' FDR control 1 Introduction Geometric median-of-means (GMOM) has been widely used for robust estimation of multivariate means, and it has been broadly adopted in machine learning (Minsker 2015, Hsu & Sabato 2016, Prasad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The idea of GMOM is to first divide the data into disjoint subsamples and calculate the empirical means of each of the subsamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then the GMOM estimator is computed as the spatial median (also called geometric median) of the obtained empirical means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The previous studies on the GMOM focused on establishing its non-asymptotic error bounds 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='03126v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ME] 9 Jan 2023 under certain heavy-tailed assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Its distributional properties, which are essential for statistical inference, remain unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' High-dimensional data with the dimension increases to infinity as the number of observa- tions goes to infinity have been encountered in many scientific disciplines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' There is a growing evidence of the multivariate normal distribution is problematic to model high-dimensional data due to the presents of heavy-tailedness and inadequate to accommodate tail dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' For example, the distributions of the microarray expression are observed to be non-normal and have heavy tails even after log transformation in many gene expression data (Purdom & Holmes 2005, Wang, Peng and Li 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' As another example, elliptical distributions, in particular the multivariate t-distribution and symmetric multivariate normal inverse Gaussian distribution, provided far superior models to the multivariate normal for daily and weekly US stock-return data (McNeil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In such cases, the sample spatial median is favored against the sample mean for estimating the location parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The above discussions strongly motivate studying the spatial median under high-dimensionality, especially its distributional properties and the implementation in statistical inference for high-dimensional location parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Let X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , Xn be a sequence of independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=') p-dimensional random vectors from a population X with cumulative distribution function FX in Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In this paper, we work on a general multivariate model where X admits the following stochastic repre- sentation: X “ θ ` νΓU , (1) where θ is the location parameter, ν is a nonnegative univariate random variable and U is a p-dimensional random vector with independent components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Model (1) covers many commonly used multivariate models and distribution families, including the independent components model (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 2015) and the elliptical distribution family (Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' We refer to Section 2 for more detailed discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Spatial median, an extension of the univariate median to multivariate distributions, was proposed for robust inference of the location parameter (Haldane 1948, Weber 1929).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The sample spatial median ˆθn P Rp minimizes the empirical criteria function Lnpβq “ řn i“1p}Xi´β}´}Xi}q, 2 where } ¨ } is the Euclidean norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Equivalently, ˆθn “ argmin βPRp Lnpβq “ argmin βPRp nÿ i“1 p}Xi ´ β} ´ }Xi}q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2) The function Lnpβq is convex, and ˆθn is unique if the observations tXiun i“1 are not concentrated on a line in Rp when p ą 2 (Milasevic & Ducharme 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' When the dimension p is fixed, the spatial median has been well studied in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' We refer to Chapter 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 of Oja (2010) for a nice review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In the high-dimensional setting, where the dimension p diverges to infinity as the number of observations n Ñ 8, there are several existing works that study the asymptotic properties of the sample spatial median.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Zou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2014) offered an expansion of ˆθn under elliptical distributions with identical shape matrix, and Cheng et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='al (2019) extended the result to a general shape matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' As a recent work, Li & Xu (2022) improved the expansion in Cheng et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='al (2019) with a smaller order remainder term under stronger conditions, and established a central limit theorem for the squared Euclidean distance }ˆθn ´θ}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In Zou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2014) and Cheng et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='al (2019), they both require that p “ Opn2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In addition, it is required in Li & Xu (2022) that p diverges at the same rate as n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' However, in modern areas such as genomics and proteomics, the dimension of the data may grow exponentially with the sample size, which lies in the “ultrahigh dimensional” region (Fan & Lv 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The previous works with restrictions on the polynomial dimensionality limit the usage of the spatial median under ultrahigh-dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Moreover, the previous results are all under elliptical distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, it is of great importance to establish asymptotic properties of the spatial median and investigate its applications under ultrahigh dimensionality and beyond elliptical distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In this paper, we first establish Gaussian and bootstrap approximations hit hyperrectangles for the sample spatial median under the general model (1) beyond elliptical distributions, which are valid when the dimension diverges exponential with the sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' They serve as the theoretical foundations of statistical inference for the location parameter based on the sample spatial median under ultrahigh dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Consistent simultaneous confidence intervals (SCIs) and global tests for the location parameters are established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' We also study multiple testing for every component of θ based on ˆθn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Motivated by simultaneous inference of θ, we 3 define a high-dimensional asymptotic relative efficiency of the sample spatial median relative to the sample mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Most importantly, our theoretical results guarantee the validity of the proposed inferential methods for exponentially divergent p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The advantages of our proposed approaches have been justified by simulations and a real data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The main contributions of this paper are summarized as follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Firstly, we establish SCIs for the location parameter θ based on the sample spatial median ˆθn, which is new in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The consistency of bootstrap approximation guarantees that the probability that the SCIs cover all components of the location parameter approaches the nominal confidence level under ultrahigh dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' We also propose a novel test for ultrahigh dimensional location parameter based on the maximum-norm of the sample spatial median.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The proposed test not only maintains nominal significance level asymptotically for exponentially divergent p, but also is more powerful under sparse alternatives compared to those based on L2-norms (Li & Xu 2022, Wang, Peng and Li 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' As another major inference, we study multiple testing for every component of the location parameter, and the false discovery rate (FDR) can be well controlled combined with the Benjamini-Hochberg procedure based on the sample spatial median, which extends the existing methods based on the sample mean (Liu and Shao 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In all inferential methods, the procedures based on the sample spatial median advances those based on the sample mean for heavy-tailed distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Secondly, this paper serves as the first work that provides Gaussian and bootstrap approxi- mations for the sample spatial median under ultrahigh dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Gaussian and bootstrap approximations for high-dimensional sample mean have received extensive attraction in the last decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2013) and Chernozhukov, Chetverikov and Kato (2017) established Gaussian and bootstrap approximations for the maxima of a sum of centered independent ran- dom vectors under Kolmogorov distance and on hyperrectangles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' See also Chen (2018), Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2019) and Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2020) for related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Compared to the sample mean, which has a simple linear form, the theoretical difficulty for the sample spatial median lies in that it does not enjoy an explicit form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' This issue is addressed by deriving a novel Bahadur representation of the sample spatial median with a maximum-norm bound on the remainder term, which extends the results of Zou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2014), Cheng et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='al (2019) and Li & Xu (2022) under elliptical distributions and polynomial dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Moreover, our results 4 can be applied to the GMOM under reasonable conditions, and thus enhance the practice usage of GMOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thirdly, we propose a novel multiplier bootstrap method for the sample spatial median.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In- stead of multiplying on the loss function, which is generally the case for M-estimator (Imaizumi & Otsu 2021), the multiplier is applied on the centralized Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Specifically, the bootstrap version of ˆθn is defined as ˜θn “ argminβPRd řn i“1 }ZipXi ´ ˆθnq ´ β}, where Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , Zn are the multipli- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The multiplier bootstrap is consistent under ultrahigh dimensionality thanks to this novel formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' This is, however, different from the multiplier bootstrap method for the sample mean, which again has an explicit form (Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 2013, Chernozhukov, Chetverikov and Kato 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Section 2 introduces model and assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Section 3 establishes Gaussian and bootstrap approximations to the distribution of the sample spatial median.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Statistical inference for the location parameter based on the sample spatial median is presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Section 5 reports numerical results including simulations and a real data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Preliminary lemmas and proofs of main results are presented in Appendix A of the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Proofs of preliminary lemmas and additional simulations are given in Appendices B and C of the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Notation: Denote |x|8 “ maxp|x1|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , |xd|q as the maximum-norm of x “ px1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , xdqJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Denote an À bn if an ď Cbn for a positive constant C, and an — bn means an À bn and bn À an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' For α ą 0, let ψαpxq “ exppxαq ´ 1 be a function defined on r0, 8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then the Orlicz norm }¨}ψα of a random variable X is defined as }X}ψα “ inf tt ą 0, Etψα p|X|{tqu ď 1u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' We use trp¨q to denote the trace operator for square matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Moreover, we denote Ip as the p ˆ p identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' For a, b P R, we write a ^ b “ minpa, bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 2 Model and assumptions In this paper, we consider a general multivariate model for the distribution FX such that Xi admits the following stochastic representation: Xi “ θ ` νiΓUi , (3) 5 where θ is the location parameter, Γ is a nonrandom and invertible p ˆ p matrix, Ui is a p- dimensional random vector with independent standardized components, and νi is a nonnegative univariate random variable independent with the spatial sign of Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The distribution of Xi depends on Γ through the shape matrix Ω “ ΓΓJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Model (3) covers many commonly used multivariate models and distribution fam- ilies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' First, the independent components model (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 2015) follows (3) with νi being a nonnegative constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Second, model (3) also includes elliptical distributions by choosing Ui „ Np0, Ipq and νi “ ξi{}Ui} for some nonnegative random variable ξi independent of Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In this case, νi is independent of the spatial sign of Ui, but not Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The independent components model has received great extension in high-dimensional data analysis as well as signal process- ing and machine learning (Hyv¨arinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In addition, the elliptical distribution family covers many non-Gaussian distributions such as multivariate t-distribution, multivariate logistic distribution, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It is commonly adopted in the literature on studying the sample spatial median (Cheng et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='al 2019, Li & Xu 2022, Zou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In terms of the GMOM, if the data are from the independent components model, the subsample means satisfy model (3) clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In addition, some subfamilies of elliptical distributions are closed under convolution, and thus the subsample means also follow model (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Our results can be applied to the GMOM estimator directly in those cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' For i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , n, and k “ 1, 2, 3, 4, denote Wi “ SpXi ´ θq and Ri “ }Xi ´ θ} (4) as the spatial-sign and radius of Xi ´ θ, where SpXq “ }X}´1XIpX ‰ 0q is the multivariate sign function with Ip¨q being the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, ˆθn satisfies řn i“1 SpXi ´ ˆθnq “ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Denote Ui “ pUi,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , Ui,pqJ, we impose the following three conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Condition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Ui,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , Ui,p are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' symmetric random variables with EpUi,jq “ 0, EpU 2 i,jq “ 1, and }Ui,j}ψα ď c0 with some constant c0 ą 0 and 1 ď α ď 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Condition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The moments ζk “ EpR´k i q for k “ 1, 2, 3, 4 exist for large enough p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In addition, there exist two positive constants b and ¯B such that b ď lim supp EpRi{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='pq´k ď ¯B for k “ 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 6 Condition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The shape matrix Ω “ pωjℓqpˆp satisfies trpΩq “ p and it belongs to the following class: Upa0ppq, m, ¯ Mq “ # Ω : m ď ωjj ď ¯ M, pÿ ℓ“1 |ωjℓ| ď a0ppq, for all j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , p + , where m ď ¯ M are bounded positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In Condition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1, the symmetric assumption is to ensure that θ in model (3) co- incides with the population spatial median, which minimizes Lpβq “ Ep}X ´ β} ´ }X}q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It is obvious that Condition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 is satisfied by elliptical distributions with Ui „ Np0, Ipq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The condi- tion }Ui,j}ψα ď c0 implies that Ui,j has a sub-exponential distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It is worth highlighting that with slight modification of the proofs of main theorems, the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' condition on Ui,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , Ui,p can be weaken by replacing Condition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 with the following assumption: Ui,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , Ui,p are in- dependent symmetric random variables with EpUi,jq “ 0, EpU 2 i,jq “ 1 for all j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , p, and sup1ďjďp }Ui,j}ψα ď c0 with some constant c0 ą 0 and 1 ď α ď 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The condition b ď lim supp EpRi{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='pq´k ď ¯B indicates that ζk — p´k{2 for k “ 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It is introduced to avoid Xi from concentrating too much near θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' For elliptical distributions, it is a generalization of Assumption 1 of Zou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2014), which is satisfied by many common distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' For the independent components model, Condition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 is equivalent to that b ď lim supp Ep}ΓUi}{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='pq´k ď ¯B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' According to Lemma A2 in Appendix A, Ep}ΓUi}kq “ pk{2t1 ` op1qu for k “ 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then the Cauchy-Schwarz inequality implies that Ep}ΓUi}´kq ě tEp}ΓUi}kqu´1 “ p´k{2t1 ` op1qu , from which we know Ep}ΓUi}´kq Á p´k{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Furthermore, denote Γj as the jth row of Γ, then by the inequality of harmonic and quadratic means, p2}ΓUi}´4 “ " p pΓ1Uiq2 ` ¨ ¨ ¨ ` pΓpUiq2 ď pΓ1Uiq´4 ` ¨ ¨ ¨ ` pΓpUiq´4 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It follows that Ep}ΓUi}´4q À p´2 if EtpΓ1Uiq´4u, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , EtpΓpUiq´4u are uniformly bounded, and from which Ep}ΓUi}´kq À p´k{2 by Jensen’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, Condition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 is satisfied by the independent components models as long as Γ1Ui, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , ΓpUi are not concentrating too much near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' See also discussions in Cardot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2013) on similar conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 7 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It is noticed that the shape matrix Ω is only well defined up to a scalar multiple, the condition trpΩq “ p is used to regularize Ω to make model (3) identifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The class Upa0ppq, m, ¯ Mq covers a wide range of symmetric square matrices, and it is commonly adopted in the literature on high-dimensional analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' For example, a similar matrix class is introduced in Bickel & Levina (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The condition m ď ωjj ď ¯ M requires bounded diagonal elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The order of a0ppq, which will be specified later, controls the orders of the off-diagonal elements of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 3 Gaussian and bootstrap approximations 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 Bahadur representation and Gaussian approximation In this section, we establish Gaussian approximation for ˆθn, which is valid when p diverges exponentially over n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The following lemma offers a Bahadur representation of ˆθn, and it severs as the foundation of the Gaussian approximation result in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (Bahadur representation) Assume Conditions C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3 with a0ppq — p1´δ for some positive constant δ ď 1{2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' If log p “ opn1{3q and log n “ opp1{3^δq, then n1{2pˆθn ´ θq “ n´1{2ζ´1 1 nÿ i“1 Wi ` Cn , where |Cn|8 “ Optn´1{4 log1{2pnpq ` p´p1{6^δ{2q log1{2pnpqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' To the best of our knowledge, Lemma 1 serves as the first result that offers the Bahadur representation of the sample spatial median with a maximum-norm bound on the remainder term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In Zou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2014) and Cheng et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='al (2019), the same expansion with the remainder term Cn satisfies }Cn} “ oppζ´1 1 q was obtained, and their result was improved to }Cn} “ opp1q in Li & Xu (2022), by replacing ζ1 with n´1 řn i“1 R´1 i in the linear term, but under a more restricted condition that p and n are of the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It is worth noticing that the previous results (Cheng et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='al 2019, Li & Xu 2022, Zou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 2014) are all derived under elliptical distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Let Are “ tśp j“1raj, bjs : ´8 ď aj ď bj ď 8, j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , pu be the class of rectangles in 8 Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' With the Bahadur representation in Lemma 1 on hand, we establish the following Gaussian approximation result for ˆθn over hyperrectangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (Gaussian approximation) Assume Conditions C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3 with a0ppq — p1´δ for some positive constant δ ď 1{2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' If log p “ opn1{5q and log n “ opp1{3^δq, then ρnpAreq “ sup APAre ˇˇˇPtn1{2pˆθn ´ θq P Au ´ P pG P Aq ˇˇˇ Ñ 0 as n Ñ 8, where G „ Np0, ζ´2 1 Bq with B “ EpW1W J 1 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The Gaussian approximation for ˆθn indicates that the probabilities Ptn1{2pˆθn ´ θq P Au can be approximated by that of a centered Gaussian random vector with covariance matrix ζ´2 1 B for hyperrectangles A P Are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Theorem 1 allows for an exponentially divergent p, which fits the ultrahigh dimensional setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Compared to the asymptotic normality of }ˆθn ´ θ}2 in Li & Xu (2022), in which p is assumed to have the same order as n, the Gaussian approximation result in Theorem 1 requires much weaker conditions on the rates of n and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Let Bjℓ be the pj, ℓqth element of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' According to Lemma A4 (iii) in Appendix A, ζ´2 1 Bjℓ “ ζ´2 1 p´1ωj,ℓ ` Opp´δ{2q for all 1 ď j, ℓ ď p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, the covariance matrix of G in Theorem 1 is asymptotically proportional to the shape matrix Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' As the sample spatial median is a special M-estimator, Gaussian approximation for M-estimator in Imaizumi & Otsu (2021) is potentially applicable to the spatial median under high-dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' However, it is worth highlighting that the results in Imaizumi & Otsu (2021) cannot be applied to our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' To be precise, Assumption 1 (A3) in Imaizumi & Otsu (2021) assumes that there exist constants C ą 0 and α P p0, 2q such that log Npε, Θ, }¨}q ď Cε´α holds for all ε P p0, 1q, where Θ is the parameter space, and Npε, Θ, } ¨ }q is the ε-covering number of Θ under the Euclidean norm } ¨ } (van der Vaart & Wellner 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' When Θ is a compact subset of Rp, Npε, Θ, } ¨ }q is of order Opε´pq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In this case, log Npε, Θ, } ¨ }q ď Cε´α cannot be satisfied when p Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, our theoretical findings are independent of those in Imaizumi & Otsu (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Theorem 1 immediately implies the following corollary since the Kolmogorov distance of sup-norm is a subset of Are corresponding to max-hyperrectangles in Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 9 Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Under the conditions assumed in Theorem 1, as n Ñ 8, ρn “ sup tPR ˇˇˇPpn1{2|ˆθn ´ θ|8 ď tq ´ Pp|G|8 ď tq ˇˇˇ Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 Multiplier bootstrap approximation Theorem 1 allows us to approximate the distribution of n1{2pˆθn ´θq by that of G hit hyperrect- angles, where G „ Np0, ζ´2 1 Bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' However, it cannot be used directly in statistical inference for θ as the quantity ζ1 and the matrix B depend on the underlying distribution FX and are thus un- known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' To solve this issue, we propose an easy-to-implement bootstrap method to approximate the distribution of n1{2pˆθn ´ θq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Let Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , Zn be a sequence of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' random variables with mean zero and unit variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Define the bootstrap version of the sample spatial median as ˜θn “ argmin βPRd nÿ i“1 }ZipXi ´ ˆθnq ´ β} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (5) Then, the distribution of n1{2˜θn conditional on X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , Xn is used to approximate that of n1{2pˆθn´θq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' This algorithm is called the multiplier bootstrap, and Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , Zn are the multiplier weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Regarding the proof of Lemma A5 in Appendix B, it is preferred that the multiplier weights Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , Zn are bounded and satisfy EpZ´2 i q ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, we choose the Rademacher variables as the multipliers (Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 2019), that is, PpZi “ 1q “ PpZi “ ´1q “ 1{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (Bootstrap approximation) Under the conditions assumed in Theorem 1, ρMB n pAreq “ sup APAre ˇˇˇPtn1{2pˆθn ´ θq P Au ´ P˚pn1{2˜θn P Aq ˇˇˇ Ñ 0 in probability as n Ñ 8, where P˚ denotes the conditional probability given X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Under the same conditions on the divergence rates of n and p as in Theorem 1, Theorem 2 validates that conditional on X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , Xn, the distribution of the bootstrap sample spatial median ˜θn approximates that of ˆθn consistently over hyperrectangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The proof of Theorem 2 is nontrivial and does not follow directly from existing 10 results since ˜θn has no explicit form, which is different from the multiplier bootstrap methods for high-dimensional sample mean that have been analysed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The key step in the proof is to obtain a Bahadur representation of ˜θn similar as ˆθn in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Specif- ically, we show that n1{2˜θn “ n´1{2ζ´1 1 řn i“1 ZiWi ` ˜Cn with | ˜Cn|8 “ Optn´1{4 log1{2pnpq ` p´p1{6^δ{2q log1{2pnpqu in Lemma A5 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The next corollary is an immediate consequence of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Under the conditions assumed in Theorem 2, as n Ñ 8, ρMB n “ sup tPR ˇˇˇPtn1{2|ˆθn ´ θ|8 ď tu ´ P˚pn1{2|˜θn|8 ď tq ˇˇˇ Ñ 0 in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 4 Statistical inference The Gaussian and multiplier bootstrap approximations for the sample spatial median enable many statistical inferential methods for ultrahigh dimensional population location parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In this section, we present the following statistical inferences: simultaneous confidence intervals (SCIs) and global tests for the population location parameter, multiple testing for every com- ponent of θ, and high-dimensional asymptotic relatively efficient of the sample spatial median compared to the sample mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 Simultaneous confidence intervals We are interested in building SCIs for all components of θ “ pθ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , θpqJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Corollary 2 motivates the following way of constructing SCIs for θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Given a nominal confidence level 1 ´ τ, define the set Cτ as Cτ “ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' θ P Rp, n1{2|ˆθn ´ θ|8 ă qB 1´τ ) , where qB 1´τ is the p1´τqth quantile of n1{2|˜θn|8 given X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Denote ˆθn “ pˆθn,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , ˆθn,pqJ, the confidence intervals are rθ´ n,j, θ` n,js for j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , p, where θ´ n,j “ ˆθn,j ´ n´1{2qB 1´τ and θ` n,j “ ˆθn,j ` n´1{2qB 1´τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 11 The next theorem shows that Cτ preserves the nominal simultaneous confidence level 1 ´ τ asymptotically under ultrahigh dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Suppose the conditions of Theorem 2 hold, then Ppθ P Cτq Ñ 1 ´ τ as n Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Equivalently, Ppθj P rθ´ n,j, θ` n,js for all 1 ď j ď pq Ñ 1 ´ τ as n Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Unlike the fixed dimensional setting, n1{2|˜θn|8 and n1{2|ˆθn ´ θ|8 are maxima of divergent numbers of variables, and their quantiles are generally divergent as p Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, Theorem 3 is not a direct consequence of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' To ascertain the consistency of Cτ theoretically, we show that, with probability approaching one, qB 1´τ is bounded by two quantiles of n1{2|ˆθn ´θ|8 with quantile levels close enough to 1´τ using an anti-concentration inequality for divergent random sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The Gaussian approximation for the sample mean ¯Xn “ n´1 řn i“1 Xi (Cher- nozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 2013, Chernozhukov, Chetverikov and Kato 2017, Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 2019) indicate that if log p “ opn1{5q, sup tPR ˇˇˇPpn1{2| ¯Xn ´ θ|8 ď tq ´ Pp|G0|8 ď tq ˇˇˇ Ñ 0 (6) as n Ñ 8 under some moderate conditions, where G0 „ Np0, Σq with Σ “ EpXXJq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Define X˚ i “ ZipXi ´ ¯Xnq for i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , n, where Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , Zn are the Rademacher weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Denote ¯X˚ n “ n´1 řn i“1 X˚ i , it has been shown in Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2019) that suptPR ˇˇPpn1{2| ¯Xn ´ θ|8 ď tq ´ P˚pn1{2| ¯X˚ n|8 ď tq ˇˇ Ñ 0 (7) in probability as n Ñ 8 when log p “ opn1{5q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Based on (7), define C1 τ “ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' θ P Rp, n1{2| ¯Xn ´ θ|8 ă qB1 1´τ ) , where qB1 1´τ is the p1 ´ τqth quantile of n1{2| ¯X˚ n|8 conditional on X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then C1 τ is also an asymptotic 1 ´ τ SCIs for θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Based on the discussion in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4, Cτ has advantage (relative shorter intervals) over C1 τ under heavy-tailed distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' We refer to Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 for finite-sample justifications on this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 Global tests for high-dimensional location parameters In this section, we propose a novel approach for global tests on high-dimensional location pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Let θ0 be a known p-dimensional vector, we are interested in testing H0 : θ “ θ0 versus H1 : θ ‰ θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (8) Theorems 1 and 2 motivate us proposing a maximum-norm type test statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Define Tn “ n1{2|ˆθn ´ θ0|8 (9) as the test statistic, and H0 is rejected when Tn is larger than a critical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' We can use the multiplier bootstrap to approximate the distribution of Tn under H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Specifically, with a nominal significance level τ, the null hypothesis is rejected if Tn ą qB 1´τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Theorem 3 guarantees that the test based on Tn maintains nominal significance level asymptotically under ultrahigh dimensionality, that is, PpTn ą qB 1´τ | H0q Ñ τ as n Ñ 8 when log p “ opn1{5q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Remark 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' An alternative test for (8) can be constructed based on ¯Xn by defining the test statistic as TMean “ n1{2| ¯Xn ´ θ0|8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then, the null hypothesis is rejected if TMean ą qB1 1´τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The test based on Tn can be deemed as a nonparametric extension of the test based on TMean .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' As ˆθn is more efficient than ¯Xn for simultaneous inference of θ under heavy-tailed distributions as discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4, we expect that the proposed test based on Tn is more powerful than that based on TMean in those cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' This has been reflected by the simulation results in Appendix C of the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The next theorem summarises the asymptotic power of the proposed test based on Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Suppose the conditions of Theorem 2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' For any given 0 ă τ ă 1, if |θ´θ0|8 ě C log1{2pτ ´1qn´1{2 log1{2pnpq for some large enough constant C ą 0, then PpTn ą qB 1´τ | H1q Ñ 1 as n Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Theorem 4 indicates that the test based on Tn achieves consistency when the maximum element of n1{2|θ ´ θ0| has a magnitude much large than log1{2pτ ´1q log1{2pnpq for a fixed significant level τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 13 Remark 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Wang, Peng and Li (2015) proposed a L2-norm type test (WPL test) for (8) with θ0 “ 0 based on TWPL “ řn i“1 ři´1 j“1 W J i Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It has been argued in Wang, Peng and Li (2015) and Li & Xu (2022) that the signal of the WPL test is determined by the magnitude of }θ}, which is the L2-norm of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' As a contrast, the power of the test based on Tn depends on |θ|8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, the proposed test based on Tn is expected to be more powerful under sparse alternatives, when θ contains only a limited number of non-zero components and its maximum element has certain order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In such cases, }θ} is not big enough for the rejection of the WPL test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' See Appendix C in the supplementary material and Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3 for numerical justifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3 Multiple testing with FDR control in large-scale tests Multiple testing with false discovery rate (FDR) control has been applied to many real problems, such as detecting differentially expressed genes in genomic study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In this section, we study multiple testing for every component of θ based on the spatial median with the Benjamini and Hochberg (B-H) method for FDR control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' For j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , p, we are interested in testing H0j : θj “ θ0,j versus H1j : θj ‰ θ0,j simultaneously, where θ0,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , θ0,p are given values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Define the test statistics as Tn,j “ n1{2pˆθn,j ´ θ0,jq{sn,j for j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , p, where s2 n,j “ ˆζ´2 1 ˆBjj with ˆζ1 “ n´1 řn i“1 }Xi ´ ˆθn}´1, and ˆBjj is the jth diagonal element of ˆB “ n´1 řn i“1 }Xi ´ ˆθn}´2pXi ´ ˆθnqpXi ´ ˆθnqJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' According to the proof of Theorem 5 in Appendix A, Tn,j converges in distribution to a standard normal under H0j for j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, we utilise the standard normal distribution to estimate the marginal p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' For j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , p, define the p-value for H0j as Pj “ 2´2Φp|Tn,j|q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Denote Pp1q ď ¨ ¨ ¨ ď Pppq be the ordered p-values, and define ˆk “ max ␣ j “ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , p : Ppjq ď τj{p ( 14 for a pre-specific significance level τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then, the B-H procedure rejects the null hypotheses for which Pj ď Ppˆkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Denote HR “ tj : Pj ď Ppˆkqu as the set of indices j such that H0j is rejected by the B-H method, and let |HR| be the cardinality of HR that equals the total number of rejected null hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Let H0 Ă t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , pu be the set of indices j corresponding to the true null hypotheses H0j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The false discovery proportion (FDP) and false discovery rate (FDR) of the B-H method are defined as FDPM “ |H0 X HR| |HR| _ 1 and FDRM “ EpFDPMq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Regarding that Tn,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , Tn,p are dependent, we impose the following condition on the weak dependence between any two components of Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Define prjℓqpˆp “ tdiagpBqu´1{2BtdiagpBqu´1{2 as the correlation matrix, where diagpBq is the diagonal matrix of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Condition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Suppose max1ďj,ℓďp |rjℓ| ď r with some constant 0 ă r ă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In addition, řp j“1 Iprjℓ “ 0q “ Oppηq for some constant 0 ă η ă p1 ´ rq{p1 ` rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Similar conditions are assumed in Liu and Shao (2014) and Belloni et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='al (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Let p0 “ |H0| be the number of true null hypotheses and Bjj be the jth diagonal element of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Suppose Condition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 and the conditions of Theorem 1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In addition, there exists H Ă t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , pu such that H “ ␣ j : ζ1B´1{2 jj n1{2|θj ´ θ0,j| ě 2 log1{2ppq ( and |H| ě log log p Ñ 8 as p Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Assume that the number of false null hypotheses p1 ď pϖ for some 0 ă ϖ ă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then, FDRM{pτp0{pq Ñ 1 as n Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Theorem 5 shows the B-H procedure based on P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , Pp controls the FDR asymptotically, and it extends Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 in Liu and Shao (2014) to spatial median-based test statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 High-dimensional asymptotic relative efficiency As two candidate estimators of the location parameter θ, it is of interest to study the asymptotic relative efficiency (ARE) of the sample spatial median ˆθn relative to the sample mean ¯Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' When p is fixed, for spherical multivariate normal distribution, Brown (1983) showed that the asymptotic efficiency of ˆθn relative ¯Xn, denoted as AREpˆθn, ¯Xnq, exceeds the usual univariate 15 case 2{π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In addition, AREpˆθn, ¯Xnq increases as the dimension increases, and it approaches to 1 as p tends to be sufficient large (Magyar & Tyler 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' However, when p Ñ 8, the ARE is not straightforward to quantify as there are no obvious “final” limit distributions for ˆθn and ¯Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Motivated by the discussions in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2, we compare ˆθn and ¯Xn in terms of their efficiencies in simultaneous inference for θ, which are determined by the variations of |ˆθ ´ θ|8 and | ¯Xn ´ θ|8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' According to Corollary 1 and (6), we define the high-dimensional ARE of ˆθn compared to ¯Xn in simultaneous inference for θ as AREpˆθn, ¯Xnq “ Varp|G0|8q{Varp|G|8q , (10) which approximates Varp| ¯Xn ´ θ|8q{Varp|ˆθn ´ θ|8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' If limpÑ8 AREpˆθn, ¯Xnq ą 1, we say that ˆθn is more efficient than ¯Xn in simultaneous inference for θ under high-dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' As discussed in Remark 6, G „ Np0, ζ´2 1 Bq with ζ´2 1 Bjℓ “ ζ´2 1 p´1ωiℓ for all 1 ď j, ℓ ď p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Moreover, we can show that Σjℓ “ Epν2 i qωjℓ ` Opp´1{2q similar to the proof of Lemma A3 in Appendix B of the supplementary material, where Σjℓ is the pj, ℓqth element of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, both the covariance matrix Σ and ζ´2 1 B are proportional to Ω asymptotically, and AREpˆθn, ¯Xnq is approximately Epν2 i qζ2 1p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' As Σ and ζ´2 1 B are rarely known in practice, we use bootstrap approximation to estimate the value of Varp|G0|8q{Varp|G|8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Combining Corollary 2 and (7), we propose using Var˚p| ¯X˚ n|8q{Var˚p|˜θn|8q, to estimate AREpˆθn, ¯Xnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Suppose X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Xn are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' from Npθ, Ipq, then ν2 i follows a chi-square distribution with p degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It follows that Epν2 i q “ p and Epν´1 i q “ Γpp{2 ´ 1{2q{t21{2Γpp{2qu, where Γp¨q is the gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='. So the ARE is AREpˆθn, ¯Xnq “ ptΓpp{2´1{2qu2{t21{2Γpp{2qu2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Using Stirling’s formula, limpÑ8 AREpˆθn, ¯Xnq “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, for high-dimensional Gaussian data, the sample spatial median has the same asymptotically efficiency as the sample mean in simul- taneous inference for θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' When the data are from the multivariate t-distribution with degrees of freedom v ą 2 and shape matrix Ω “ Ip, ν2 i {p „ Fp,v, where Fp,v is the F distribution with parameters 16 p and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then, Epν2 i q “ pv{pv ´ 2q and Epν´1 i q “ Γpv{2 ` 1{2qΓpp{2 ´ 1{2q{tv1{2Γpv{2qΓpp{2qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, the ARE is AREpˆθn, ¯Xnq “ pv ´ 2q´1ptΓpv{2 ` 1{2qΓpp{2 ´ 1{2qu2{tΓpv{2qΓpp{2qu2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It is clear that AREpˆθn, ¯Xq ą 1 for large enough p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In addition, limpÑ8 AREpˆθn, ¯Xnq “ 2pv ´ 2q´1tΓpv{2 ` 1{2qu2{tΓpv{2qu2 ą 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, for high-dimensional t-distribution, the sample spatial median is asymptotically more efficient than the sample mean in simultaneous inference for θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Figure 1 plots the simulated values of Varp| ¯Xn|8q{Varp|ˆθn|8q with a range of dimensions and sample sizes under different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' For Gaussian data, the relative efficiency kept increas- ing in p, and it approached 1 as p getting larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' For the data simulated from multivariate t-distribution, the relative efficiency was greater than 1 for all combinations of n and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' This in- dicates that the sample spatial median is more efficiency than the sample mean for t-distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The results were consistent under different covariance structure considered in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 100 200 300 400 p Efficiencies Gaussian, ρ=0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 0 100 200 300 400 p Efficiencies t5, ρ=0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 100 200 300 400 p Efficiencies Gaussian, ρ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='9 0 100 200 300 400 p Efficiencies t5, ρ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 n 20 40 80 Figure 1: Finite sample relative efficiency of |ˆθn|8 compared to | ¯Xn|8 based on 5000 replications, the data are generated from multivariate normal distribution (Gaussian) and t-distribution with 5 degrees of freedom (t5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The shape matrix Ω “ pρ|j´ℓ|qpˆp with ρ “ 0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 17 5 Numerical studies In this section, we report Monte Carlo simulations on simultaneous confidence intervals and multiple testing with FDR control, along with a real data analysis, to demonstrate the per- formance of the proposed approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Additional simulations on global tests can be found in Appendix C of the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In the simulations, all results were based on 2500 replications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In the bootstrap implementation, the number of bootstrap iterations was set to B “ 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 Simulations on simultaneous confidence intervals We first examine the performance of the SCIs based on ˆθn, and compare it with the SCIs based on ¯Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The sample size n is taken to be 100 or 200, and the dimensions p “ 100 and 1000 are considered for each sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Two types of commonly used elliptical distributions are considered: (I) the multivariate normal distribution Npθ, Σq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (II) the multivariate t-distribution with 3 degrees of freedom, mean vector θ, and covariance matrix Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In addition, we include the following independent components model: (III) Xi “ θ ` Σ1{2Zi, where each component of Zi are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' from the standard Laplace distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' We set Σ “ pρ|j´ℓ|q with ρ “ 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' To save space, we present the results for ρ “ 0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The results for ρ P t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5u are similar and are reported in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' We consider both sparse and dense case scenarios for θ: (i) θ1 “ p2, ´2, 3, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , 0q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (ii) θ2 “ p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2tp{4u, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , 0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Here t¨u is the floor function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Table 1 reports the coverage probability and median length of the SCIs based on ˆθn, the results of the SCIs based on ¯Xn are presented in parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' For Models I and II from elliptical distributions, we observe that the SCIs based on ˆθn and ¯Xn both achieve satisfying coverage probability for different choices for ρ, θ, n and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' For the data simulated from the multivariate normal distribution, the median length of the SCIs based on ˆθn is very close to that of the the SCIs based on ¯Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' These results indicate that ˆθn has similar asymptotic efficiency as ¯Xn in simultaneous inference for θ under high-dimensional Gaussian model as discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' For the multivariate t-distribution, the SCIs based on ˆθn is much narrower than the SCIs based on ¯Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' These results suggest that the SCIs based on ˆθn is more efficient than the SCIs 18 based on ¯Xn for multivariate t-distribution, which is heavy-tailed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' This is consistent with the asymptotic analysis in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Moreover, the results for Model III, which does not belong to the elliptical distribution family, shows the robustness of the SCIs based on the spatial median, and it performs similar to the SCIs based on the sample mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' We also note that the median length of the SCIs decreases when n increases or p decreases for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Table 1: Coverage probability (in %) and median length of the SCIs based on ˆθn, the results of the SCIs based on ¯Xn are in parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' θ “ θ1 θ “ θ2 Coverage probability Median length Coverage probability Median length Model ρ n p 90% 95% 90% 95% 90% 95% 90% 95% I 0 100 100 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='9) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='65 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='65) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='69 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='69) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='9 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 (93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='9) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='65 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='65) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='69 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='69) 1000 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='77 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='77) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='80 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='80) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='77 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='77) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='81 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='80) 200 100 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='46 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='46) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='49 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='49) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='46 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='46) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='49 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='49) 1000 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='55 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='55) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='57 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='57) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='55 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='55) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='57 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='57) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 100 100 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='64 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='63) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='68 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='67) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='64 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='63) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='68 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='67) 1000 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 (93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='76 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='76) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='80 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='79) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='76 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='76) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='80 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='79) 200 100 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 (90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='9) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='45 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='45) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='48 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='48) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='45 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='45) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='48 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='48) 1000 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='54 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='54) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='56 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='56) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 (93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='54 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='54) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='56 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='56) II 0 100 100 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7 (93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='71 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='05) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='11) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='71 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='05) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='11) 1000 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='84 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='88 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='30) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='84 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='88 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='31) 200 100 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='76) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='53 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='81) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='76) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='53 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='80) 1000 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='59 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='90) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='62 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='94) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 (93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='9) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='59 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='90) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='62 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='94) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 100 100 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 (90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='9) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='69 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='02) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='74 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='09) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='69 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='02) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='74 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='09) 1000 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='83 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='23) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='87 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='29) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='83 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='23) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='87 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='29) 200 100 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 (87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='49 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='73) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='52 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='78) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3 (90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='9 (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='49 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='73) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='52 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='78) 1000 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='59 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='88) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='61 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='92) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 (90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7 (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='59 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='89) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='61 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='93) III 0 100 100 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='65 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='66) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='69 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='70) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='65 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='66) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='69 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='70) 1000 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 (93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='78 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='78) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='82 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='82) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 (93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='78 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='78) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='82 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='82) 200 100 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 (91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='46 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='46) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='49 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='49) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 (90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='46 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='46) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='49 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='49) 1000 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 (90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='55 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='55) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='57 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='58) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7 (89) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 (93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='55 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='55) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='57 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='58) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 100 100 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='63 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='63) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='68 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='68) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='9) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='9) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='63 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='63) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='67 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='68) 1000 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='9) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='77 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='77) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='80 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='80) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='76 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='76) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='80 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='80) 200 100 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='45 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='45) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='48 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='48) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='45 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='45) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='48 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='48) 1000 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='54 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='54) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='57 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='57) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='54 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='54) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='57 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='57) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 Simulations on multiple testing with FDR control In this section, we examine the performance of the sample spatial median-based B-H method introduced in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3, and compare it to the B-H procedure based on the sample mean with p-values calculated from Np0, 1q in Liu and Shao (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' We set θ0,j “ 0 for all j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The data are generated from Models I and II with p “ 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' For θ “ pθ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , θpqJ, let θj “ 2plog p{nq1{2 for 1 ď j ď p1 and θj “ 0 for pp1 ` 1q ď j ď p, where p1 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Table 2 reports the empirical FDR and power for the sample spatial median-based (FDRM and powerM) and the sample mean-based (FDRA and powerA) B-H procedures (Liu and Shao 19 2014) with nominal level α “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The results indicate that the FDR are well controlled by both methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' For the multivariate normal distribution, the B-H procedures based on the spatial median and the sample mean have similar performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' However, the sample spatial median-based B-H method outperforms the sample mean-based B-H procedure in terms of empirical power under multivariate t-distribution, which is heavy-tailed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Table 2: Empirical FDR and power for the spatial median-based (FDRM and powerM) and the sample mean-based (FDRA and powerA) in Liu and Shao (2014) via B-H procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' α “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 α “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 Model ρ n FDRM FDRA powerM powerA FDRM FDRA powerM powerA I 0 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='224 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='222 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='999 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='202 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='201 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='224 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='223 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='999 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='202 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='201 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='223 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='999 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='202 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='201 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='127 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='227 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='223 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='999 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='204 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='199 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='999 II 0 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='117 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='984 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='728 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='193 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='805 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='088 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='987 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='710 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='197 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='179 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='795 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='117 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='098 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='984 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='727 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='194 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='805 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='087 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='987 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='709 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='198 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='179 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='795 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='984 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='727 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='216 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='194 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='803 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='087 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='987 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='708 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='198 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='178 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='794 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='098 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='984 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='724 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='218 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='192 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='800 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='087 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='987 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='705 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='199 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='177 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='791 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3 Real data analysis Type 2 diabetesis a disease in which the body becomes resistant to normal effects of insulin and gradually loses the capacity to produce enough insulin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Because skeletal muscle is the main tissue for insulin-stimulated glucose disposal, skeletal muscle insulin resistance is com- monly viewed as the critical component of whole-body insulin resistance, and thus is crit- ical to the pathogenesis of Type 2 diabetes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' To investigate the effects of insulin on gene expression in skeletal muscle, a microarray study was performed in 15 diabetic patients us- ing the Affymetrix Hu95A chip of muscle biopsies both before and after insulin treatment (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In this paper, we are interested in the gene expression alteration, that 20 is, the change of the gene expression level, due to the treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The data are available at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ncbi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='nlm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='nih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='gov/geo/query/acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='cgi?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='acc=GSE22309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The data were normal- ized by quantile normalization by the normalizeQuantiles function in the limma R package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Follow Wang, Peng and Li (2015), we focused on 2547 curated gene sets with at least 15 genes, which are from the C2 collection of the GSEA online pathway databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The gene expression values are consolidated by taking the average when multiple probes are associated with the same gene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' We implemented the Median test based on Tn on the 2519 gene sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' This is equivalent to testing whether the median change vector of gene expression levels is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The number of bootstrap iterations is B “ 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' With the Bonferroni correction, there are 1242 gene sets identified as significant at 5% level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' For comparison, we applied the WPL test (Wang, Peng and Li 2015) and the CQ test (Chen and Qin 2010) on the same gene sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' For the WPL test, 1060 gene sets are selected as significant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' and for the CQ test, 630 gene sets are identified as significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Out of the 630 gene sets selected by the CQ test, 605 of them are also identified by our proposed method, and 629 of them are identified by the WPL test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It has been argued in Wang, Peng and Li (2015) that some gene expression levels have heavy tails as their kurtosises are much larger than the kurtosis of a normal distribution, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, the methods based on the spatial median (Median test and the WPL test) are expected to be more robust and efficient than those based on moments (CQ test).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In addition, out of the 1060 gene sets identified by the WPL test, 958 of them are significant based on our proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' As argued in Remark 12, the Median test based on Tn is more powerful in detecting strong sparse signal compared to the WPL test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' To see this, we look into the following three gene sets: (1) ZHAN MULTIPLE MYELOMA UP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2) MIKKELSEN MEF HCP WITH H3K27ME3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (3) JAZAG TGFB1 SIGNALING VIA SMAD4 UP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The p-values of the WPL test for these three gene sets are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='41, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='31, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='27, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' However, the p-values of the Median test are all less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 ˆ 10´5 with B “ 105 bootstrap iterations for these three gene sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Figure 2 plots the SCIs for the spatial median vectors of the change of gene expression levels for these three gene sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The confidence intervals that do not cover 0 are colored in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It is very clear that the only one or two big values in the spatial median 21 results in a rejection of the Median test, while the signals from other dimensions are not strong enough to land a rejection by the the WPL test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Finally, we use the spatial median-based B-H procedure to perform multiple testing with FDR control on the three gene sets to detect differentially expressed genes (DEG), which is one of the most important targets in genomic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Table 3 reports the detected differentially expressed genes (DEG) in each gene set with nominal level α “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1, along with the corresponding marginal p-value Pj “ 2 ´ 2Φp|Tn,j|q and the confidence interval in the SCIs for the selected genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It can be seen that for all the selected genes, the marginal p-values are very small, and the corresponding confidence intervals do not cover 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Table 3: Detected differentially expressed genes (DEG) by the spatial median-based B-H proce- dure for three gene sets with α “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' “p-value” refers to the marginal p-value Pj “ 2´2Φp|Tn,j|q, and “CI” refers to the confidence interval in the SCIs for the selected genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Gene set DEG p-value CI ZHAN MULTIPLE MYELOMA UP CDKN1A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00082 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='234, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='550) MIKKELSEN MEF HCP WITH H3K27ME3 MYOD1 ă 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='433, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='791) JAZAG TGFB1 SIGNALING VIA SMAD4 UP HDAC4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00058 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='254, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='644) 6 Discussion In this paper, we established one-sample and two-sample Gaussian and bootstrap approxima- tions for ultrahigh dimensional sample spatial median under a general model beyond elliptical distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It is of interest to study whether our results are potentially extendable to some other distribution families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' We leave this to a future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In addition, the proposed test based on the maxima of the sample spatial median is more powerful under sparse alternatives compared to those based on L2-norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It is well known that the L2-norm type tests are more powerful under dense alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, it is of interest to consider combining the test based on the maximum-norm and L2-norm, which could be potentially powerful under both sparse and dense alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' We also leave this to a future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 22 ZHAN_MULTIPLE_MYELOMA_UP 0 20 40 60 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 Gene Index Spatial Median MIKKELSEN_MEF_HCP_WITH_H3K27ME3 0 100 200 300 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 Gene Index Spatial Median JAZAG_TGFB1_SIGNALING_VIA_SMAD4_UP 0 20 40 60 80 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 Gene Index Spatial Median Figure 2: Simultaneous Confidence intervals (SCIs) for spatial medians of three gene sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 23 Supplementary material The supplementary material includes all the technical proofs and some additional numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' References Belloni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Chernozhukov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Chetverikov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Hansen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' & Kato, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2018), ‘High- dimensional econometrics and generalized gmm’, arXiv p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='01888.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Bickel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' & Levina, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2008), ‘Covariance regularization by thresholding’, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Statist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 36, 2577–2604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Brown, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (1983), ‘Statistical uses of the spatial median’, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Statist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' B 45, 25–30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Cardot, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', C´enac, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' & Zitt, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2013), ‘Efficient and fast estimation of the geometric median in hilbert spaces with an averaged stochastic gradient algorithm’, Bernoulli 19, 18–43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' & Qin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2010), ‘A two-sample test for high-dimensional data with applications to gene-set testing’, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Statist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 38, 808–835.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2018), ‘Gaussian and bootstrap approximations for high-dimensional U-statistics and their applications’, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Statist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 46, 642–678.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Cheng, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Liu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Peng, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Zhang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' & Zheng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2019), ‘Testing the equality of two high- dimensional spatial sign covariance matrices’, Scand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Statist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 46, 257–271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Chernozhukov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Chetverikov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' & Kato, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2013), ‘Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors’, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Statist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 41, 2786– 2819.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Chernozhukov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Chetverikov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' & Kato, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2017), ‘Central limit theorems and bootstrap in high dimensions’, The Annals of Probability 45, 2309–2352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Chernozhukov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Chetverikov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' & Kato, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2019), ‘Improved central limit theorem and bootstrap approximation in high dimensions’, arXiv p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='10529.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 24 Chernozhukov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Chetverikov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Kato, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' & Koike, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2020), ‘Nearly optimal central limit theorem and bootstrap approximations in high dimensions’, arXiv p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='09513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Fan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' & Lv, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2008), ‘Sure independence screening for ultrahigh dimensional feature space’, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Statist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' B 70, 849–911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Fang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Kotz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' & Ng, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (1990), Symmetric multivariate and related distributions, Boca Raton, FL: CRC Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Haldane, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (1948), ‘Note on the median of a multivariate distribution’, Biometrika 35, 414– 417.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Hsu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' & Sabato, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2016), ‘Loss minimization and parameter estimation with heavy tails’, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 17, 1–40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Hyv¨arinen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Karhunen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' & Oja, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2001), Independent Component analysis, New York: Wiley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Imaizumi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' & Otsu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2021), ‘On gaussian approximation for m-estimator’, arXiv p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='15678v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Koike, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2021), ‘Notes on the dimension dependence in high-dimensional central limit theorems for hyperrectangles’, Japanese Journal of Statistics and Data Science 1, 257–297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' & Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2022), ‘Asymptotic properties of high-dimensional spatial median in elliptical distributions with application’, Journal of Multivariate Analysis 190, 104975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Liu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' & Shao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2014), ‘Phase transition and regularized bootstrap in large scale t-tests with false discovery rate control’, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Statist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 42, 2003–2025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Magyar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' & Tyler, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2011), ‘The asymptotic efficiency of the spatial median for elliptically symmetric distributions’, Sankhya B 73, 165–192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' McNeil, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Frey, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' & Embrechts, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2005), Quantitative Risk Management: Concepts, Techniques and Tools, Princeton, NJ: Princeton University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Milasevic, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' & Ducharme, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (1987), ‘Uniqueness of the spatial median’, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Statist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 15, 1332–1333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 25 Minsker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2015), ‘Geometric median and robust estimation in banach spaces’, Bernoulli 21, 2308–2335.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Oja, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2010), Multivariate nonparametric methods with R: An approach based on spatial signs and ranks, Lecture Notes in Statistics, Springer, New York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Prasad, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Suggala, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Balakrishnan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' & Ravikumar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2020), ‘Robust estimation via robust gradient estimation’, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Statist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' B 82, 601–627.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Purdom, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' & Holmes, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2005), ‘Error distribution for gene expression data’, Statistical Applications in Genetics and Molecular Biology 4, 1–35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' van der Vaart, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' & Wellner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (1996), Weak Convergence and Empirical Processes: With Applications to Statistics, Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Peng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' & Li, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2015), ‘A high-dimensional nonparametric multivariate test for mean vector’, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Statist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 110, 1658–1669.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Weber, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (1929), Uber Den Standort der Industrien (Alfred Weber?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='s Theory of the Location of Industries), Chicago, IL: Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Chicago Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Wu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Cui, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Maianu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Rhees, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Rosinski, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', So, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Willi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Osier, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Hill, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Page, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Allison, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Maritin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' & Garvey, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2007), ‘The effect of insulin on expression of genes and biochemical pathways in human skeletal muscle’, Endocrine 31, 5–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Yao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Zheng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' & Bai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2015), Sample covariance matrices and high-dimensional data analysis, Cambridge University Press, Cambridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Zou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Peng, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Feng, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' & Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2014), ‘Multivariate sign-based high-dimensional tests for sphericity’, Biometrika 101, 229–236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 26 Supplement to “Statistical Inference for Ultrahigh Dimensional Location Parameter Based on Spatial Median” Appendix A: Technical Proofs We first introduce and recall some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' For a d1 ˆ d2 matrix M “ pmjℓqd1ˆd2, its matrix ϱ-norm is }M}ϱ “ supt}Mx}ϱ : }x}ϱ “ 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Specifically, the 1-, 2-, and 8-norms of M are }M}1 “ max1ďℓďd2 řd1 j“1 |mjℓ|, }M}2 “ tλmaxpMJMqu1{2, and }M}8 “ max1ďjďd1 řd2 ℓ“1 |mjℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The Frobenius norm of M is }M}F “ třd1 j“1 řd2 ℓ“1 m2 jℓu1{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Define a random p ˆ p matrix Q “ n´1 řn i“1 R´1 i WiW J i such that EpQq “ EpR´1 i WiW J i q, and denote Qjℓ as the pj, ℓqth element of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Denote E˚p¨q and Var˚p¨q be the expectation and variance conditional on X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , Xn, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Recall that Wi,j is the jth element of Wi for i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , n and j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' ωjℓ is the pj, ℓqth element of Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' and Γj is the jth row of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Finally, we will denote various positive absolute constants by C1, C2, C3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' without mentioning this explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 Preliminary lemmas In this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' we present several preliminary lemmas, whose proof can be found in online Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Lemma A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (Concentration of norms) Suppose that Conditions C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3 hold with a0ppq — p1´δ for some positive constant δ ď 1{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then, for sufficient large p, there exist positive con- stants c1 and c2 such that P !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' p ´ ϵpp1`δq{2 ď }U1}2 ď p ` ϵpp1`δq{2) ě 1 ´ c1 expt´c2pδα{p4α`4qu and P ␣ p1 ´ ϵqtrpΩq ď }ΓU1}2 ď p1 ` ϵqtrpΩq ( ě 1 ´ c1 expt´c2pδα{p4α`4qu 27 for any fixed 0 ă ϵ ă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Lemma A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Suppose that Conditions C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3 hold with a0ppq — p1´δ for some positive constant δ ď 1{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then, for any i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , n, (i) Ep}Ui}4q “ pEpU 4 i,jq ` ppp ´ 1q, Ep}Ui}6q “ pEpU 6 i,jq ` 3ppp ´ 1qEpU 4 i,jq ` ppp ´ 1qpp ´ 2q, Ep}Ui}8q “ pEpU 8 i,jq ` 4ppp ´ 1qEpU 6 i,j1q ` 3ppp ´ 1qtEpU 4 i,j1qu2 `3ppp ´ 1qEpU 4 i,jq ` ppp ´ 1qpp ´ 2qpp ´ 3q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In addition, Ep}Ui}2kq “ pk ` Oppk´1q and Ep}U}kq “ pk{2 ` Oppk{2´1q for any positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (ii) Ep}ΓUi}4q “ p2 ` Opp2´δq, Ep}ΓUi}6q “ p3 ` Opp3´δq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In addition, Ep}ΓUi}q “ p1{2 ` Opp1{2´δq and Ep}ΓUi}3q “ p3{2 ` Opp3{2´δq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (iii) Et}ΓSpUiq}2u “ 1 ` Opp´1{2q and Et}ΓSpUiq}4u “ 1 ` Opp´1{3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (iv) Epν´k i q À ζkpk{2 for k “ 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Lemma A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Suppose Conditions C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3 with a0ppq — p1´δ for some positive constant δ ď 1{2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Define a random pˆp matrix Q “ n´1 řn i“1 R´1 i WiW J i and let Qjℓ be the pj, ℓqth element of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then, (i) |Qjℓ| À ζ1p´1|ωjℓ| ` Oppζ1n´1{2p´1 ` ζ1p´7{6 ` ζ1p´1´δ{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (ii) Qjℓ “ Q0,jℓ ` Oppζ1p´7{6 ` ζ1p´1´δ{2q, where Q0,jℓ is the pj, ℓqth element of Q0 “ n´1p´1{2 nÿ i“1 ν´1 i tΓSpUiqutΓSpUiquJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In addition, Q0 satisfies trrEpQ2 0q ´ tEpQ0qu2s “ Opn´1p´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Lemma A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Suppose Conditions C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3 with a0ppq — p1´δ for some positive constant δ ď 1{2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then, (i) Etpζ´1 1 Wi,jq4u À ¯ M2 and Etpζ´1 1 Wi,jq2u Á m for all i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , n and j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (ii) }ζ´1 1 Wi,j}ψα À ¯B for all i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , n and j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 28 (iii) EpW 2 i,jq “ p´1ωjj ` Opp´1´δ{2q for j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , p and EpW 2 i,jq “ p´1ωjℓ ` Opp´1´δ{2q for 1 ď j ‰ ℓ ď p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (iv) if log p “ opn1{3q, ˇˇˇˇˇn´1{2 nÿ i“1 ζ´1 1 Wi ˇˇˇˇˇ 8 “ Optlog1{2pnpqu and ˇˇˇˇˇn´1 nÿ i“1 pζ´1 1 Wiq2 ˇˇˇˇˇ 8 “ Opp1q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Lemma A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Suppose the conditions of Theorem 2 hold, then n1{2˜θn “ n´1{2ζ´1 1 nÿ i“1 ZiWi ` ˜Cn , (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='11) where | ˜Cn|8 “ Optn´1{4 log1{2pnpq ` p´p1{6^δ{2q log1{2pnpqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The following lemma is Nazarov’s inequality, and its proof can be found in Chernozhukov, Chetverikov and Kato (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Lemma A6 (Nazarov’s inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Let Y0 “ pY0,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , Y0,pqJ be a centered Gaussian random vector in Rp and EpY 2 0,jq ě b for all j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , p and some constant b ą 0, then for every y P Rp and a ą 0, PpY0 ď y ` aq ´ PpY0 ď yq À a log1{2ppq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 Proof of main results Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' As θ is a location parameter, we assume θ “ 0 without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then, Wi “ SpXiq “ }Xi}´1Xi “ }ΓUi}´1ΓUi for i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The sample spatial median ˆθn satisfies nÿ i“1 SpXi ´ ˆθnq “ nÿ i“1 Xi ´ ˆθn }Xi ´ ˆθn} “ nÿ i“1 Wi ´ R´1 i ˆθn }Wi ´ R´1 i ˆθn} “ 0 , which is is equivalent to n´1 nÿ i“1 pWi ´ R´1 i ˆθnqp1 ´ 2R´1 i W J i ˆθn ` R´2 i }ˆθn}2q´1{2 “ 0 as W J i Wi “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Under Condition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2, ζk “ EpR´k i q “ Opp´k{2q for k “ 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In addition, Lemma 29 A3 indicates that Qjℓ “ Q0,jℓ ` Oppζ1p´7{6 ` ζ1p´1´δ{2q, where Q0,jℓ is the pj, ℓqth element of Q0 “ n´1p´1{2 řn i“1 νitΓSpUiqutΓSpUiquJ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In addition, Q0 satisfies trrEpQ2 0q ´ tEpQ0qu2s “ Opn´1p´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, from the similar procedure as in the proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 of Cheng et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='al (2019), we can show that }ˆθn} “ Oppζ´1 1 n´1{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then, for i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , n, we have |R´1 i W T i ˆθn| ď R´1 i }ˆθn} “ Oppn´1{2q and R´2 i ||ˆθn||2 “ Oppn´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' By the first-order Taylor expansion, the above equation can be rewritten as n´1 nÿ i“1 pWi ´ R´1 i ˆθnqp1 ` R´1 i W J i ˆθn ´ 2´1R´2 i }ˆθn}2 ` δ1iq “ 0 , (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='12) where δ1i “ OptpR´1 i W J i ˆθn ´ 2´1R´2 i }ˆθn}2q2u “ Oppn´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' By Markov’s inequality, for any ε ą 0, P ˆ max 1ďiďn R´1 i ě εζ1n1{4 ˙ “ P ˆ max 1ďiďn R´4 i ě ε4ζ4 1n ˙ ď E ˆ max 1ďiďn R´4 i ˙ {pε4ζ4 1nq ď nEpR´4 i q{pε4ζ4 1nq À ε´4 , where the last inequality is due to Condition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, max1ďiďn R´2 i “ Oppζ2 1n1{2q, and consequently, max1ďiďn δ1i “ Opp}ˆθn}2 max1ďiďn R´2 i q “ Oppn´1{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Rewrite (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='12) as n´1 nÿ i“1 p1 ´ 2´1R´2 i }ˆθn}2 ` δ1iqWi ` n´1 nÿ i“1 R´1 i pW J i ˆθnqWi “ n´1 nÿ i“1 R´1 i p1 ´ 2´1R´2 i }ˆθn}2 ` δ1iqˆθn ` n´1 nÿ i“1 R´2 i pW J i ˆθnqˆθn , which implies n´1 nÿ i“1 p1 ´ 2´1R´2 i }ˆθn}2 ` δ1iqWi ` n´1 nÿ i“1 R´1 i pW J i ˆθnqWi (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='13) “ n´1 nÿ i“1 R´1 i p1 ` δ1i ` δ2iqˆθn , where δ2i “ R´1 i W J i ˆθn ´ 2´1R´2 i }ˆθn}2 “ Oppδ1{2 1i q satisfies max1ďiďn δ2i “ Oppn´1{4q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It is 30 straightforward to check that n´1 řn i“1 R´1 i pW J i ˆθnqWi “ n´1 řn i“1 R´1 i WiW J i ˆθn “ Qˆθn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' From Lemma A3, |Qjℓ| À ζ1p´1|ωjℓ| ` Oppζ1n´1{2p´1 ` ζ1p´7{6 ` ζ1p´1´δ{2q, and this implies that |Qˆθn|8 ď }Q}1||ˆθn|8 À ζ1p´1}Ω}1|ˆθn|8 ` Oppζ1n´1{2 ` ζ1p´1{6 ` ζ1p´δ{2q|ˆθn|8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' According to Lemma A4, we have that |n´1 řn i“1 ζ´1 1 Wi|8 “ Optn´1{2 log1{2pnpqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then, ˇˇˇˇˇζ´1 1 n´1 nÿ i“1 δ1iWi ˇˇˇˇˇ 2 8 ď ˇˇˇˇˇn´1 nÿ i“1 pζ´1 1 Wiq2 ˇˇˇˇˇ 8 ˜ n´1 nÿ i“1 δ2 1i ¸ À Oppn´2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In addition, we have that |ζ´1 1 n´1 řn i“1 R´2 i }ˆθn}2Wi|8 À Oppn´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Regarding equation (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='13) and the fact that ζ´1 1 n´1 řn i“1 R´1 i “ 1 ` Oppn´1{2q , we obtain ˆθn|8 À ˇˇˇˇˇζ´1 1 n´1 nÿ i“1 Wi ˇˇˇˇˇ 8 ` ζ´1 1 |Qˆθn|8 À p´1a0ppq|ˆθn|8 ` Oppn´1{2 ` p´p1{6^δ{2qq|ˆθn|8 ` Optn´1{2 log1{2pnpqu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, we conclude that |ˆθn|8 “ Optn´1{2 log1{2pnpqu as a0ppq — p1´δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In addition, we have |ζ´1 1 Qˆθn|8 “ Optn´1{2p´p1{6^δ{2q log1{2pnpq`n´1 log1{2pnpqu and n´1 řn i“1 R´1 i p1`δ1i `δ2iq “ ζ1t1 ` Oppn´1{4qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Finally, we can write n1{2pˆθn ´ θq “ n´1{2ζ´1 1 nÿ i“1 Wi ` Cn , where Cn satisfies |Cn|8 “ Optn´1{4 log1{2pnpq ` p´p1{6^δ{2q log1{2pnpqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Let Ln,p “ n´1{4 log1{2pnpq`p´p1{6^δ{2q log1{2pnpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then, for any sequence ηn Ñ 8 and any t P Rp, P !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' n1{2pˆθn ´ θq ď t ) “ P ˜ n´1{2ζ´1 1 nÿ i“1 Wi ` Cn ď t ¸ 31 ď P ˜ n´1{2ζ´1 1 nÿ i“1 Wi ď t ` ηnLn,p ¸ ` Pp|Cn|8 ą ηnLn,pq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' According to Lemma A4, Etpζ´1 1 Wi,jq4u À ¯ M2, Etpζ´1 1 Wi,jq2u Á m, and }ζ´1 1 Wi,j}ψα À ¯B for all i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , n and j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' According to the Gaussian approximation for independent partial sums in Koike (2021), let G „ Np0, ζ´2 1 Bq with B “ EpW1W J 1 q, we have P ˜ n´1{2ζ´1 1 nÿ i“1 Wi ď t ` ηnLn,p ¸ ď PpG ď t ` ηnLnpq ` O ´␣ n´1 log5pnpq (1{6¯ ď PpG ď tq ` OtηnLnp log1{2ppqu ` O ´␣ n´1 log5pnpq (1{6¯ , where the last inequality is from Nazarov’s inequality in Lemma A6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It is also worth noting that the order O ´␣ n´1 log5pnpq (1{6¯ is improved to O ´␣ n´1 log5pnpq (1{4¯ in Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, P !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' n1{2pˆθn ´ θq ď t ) ď PpG ď tq ` OtηnLnp log1{2ppqu ` O ´␣ n´1 log5pnpq (1{6¯ `Pp|Cn|8 ą ηnLn,pq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' On the other hand, we also have P !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' n1{2pˆθn ´ θq ď t ) ě PpG ď tq ´ OtηnLnp log1{2ppqu ´ O ´␣ n´1 log5pnpq (1{6¯ ´Pp|Cn|8 ą ηnLn,pq , where Pp|Cn|8 ą ηnLn,pq Ñ 0 as n Ñ 8 according to Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then, if log p “ opn1{5q and log n “ opp1{3^δq, with sufficiently slow ηn Ñ 8, we have sup tPRp ˇˇˇPtn1{2pˆθn ´ θq ď tu ´ PpG ď tq ˇˇˇ Ñ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 32 We obtain immediately from Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 in Chernozhukov, Chetverikov and Kato (2017) that ρnpAreq “ sup APAre ˇˇˇPtn1{2pˆθn ´ θq P Au ´ P pG P Aq ˇˇˇ Ñ 0 , which leads to the conclusion of this theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Let ˜Xi “ Xi ´ ˆθn and ˜Ri “ } ˜Xi} for i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' According to Lemma A5, n1{2˜θn “ n´1{2ζ´1 1 nÿ i“1 ZiWi ` ˜Cn , where ˜Cn satisfies | ˜Cn|8 “ Optn´1{4 log1{2pnpq ` p´p1{6^δ{2q log1{2pnpqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Denote ¯Wn “ n´1 řn i“1 Wi and rewrite n1{2˜θn “ n´1{2ζ´1 1 nÿ i“1 ZipWi ´ ¯Wnq ` ˜ n´1{2ζ´1 1 nÿ i“1 Zi ¸ ¯Wn ` ˜Cn, where ˇˇˇˇˇ ˜ n´1{2ζ´1 1 nÿ i“1 Zi ¸ ¯Wn ˇˇˇˇˇ 8 ď ζ´1 1 ˇˇˇˇˇn´1{2 nÿ i“1 Zi ˇˇˇˇˇ ˇˇ ¯Wn ˇˇ 8 À n´1{2 log1{2pnpq according to Lemma A4 (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It is clear that E˚ ␣ n´1{2ζ´1 1 řn i“1 ZipWi ´ ¯Wnq ( “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Let ˆB “ n´1 řn i“1 WiW J i , then Var˚ # n´1{2ζ´1 1 nÿ i“1 ZipWi ´ ¯Wnq + “ ζ´1 1 ˆB ´ ζ´2 1 ¯Wn ¯W J n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Denote Bjℓ and ˆBjℓ be the pj, ℓqth element of B and ˆB, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In addition, denote ¯Wn,j as the jth element of ¯Wn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Define ∆n “ max 1ďj,ℓďp ˇˇˇζ´2 1 ˆBjℓ ´ ζ´2 1 ¯Wn,j ¯Wn,ℓ ´ ζ´2 1 Bjℓ ˇˇˇ , 33 then ∆n ď ∆n1 ` max 1ďj,ℓďp ˇˇζ´2 1 ¯Wn,j ¯Wn,ℓ ˇˇ À ∆n1 ` n´1 logpnpq, where ∆n1 “ max 1ďj,ℓďp |ζ´2 1 ˆBjℓ ´ ζ´2 1 Bjℓ| “ max 1ďj,ℓďp ˇˇˇˇˇn´1ζ´2 1 nÿ i“1 tWi,jWi,ℓ ´ EpWi,jWi,ℓqu ˇˇˇˇˇ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' From the properties of the ψα norm, it holds that ›››› max 1ďiďn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1ďj,ℓďp |ζ´2 1 Wi,jWi,ℓ| ›››› ψα{2 À ›››› max 1ďiďn,1ďjďp |ζ´2 1 Wi,j|2 ›››› ψα{2 “ ζ´2 1 ›››› max 1ďiďn,1ďjďp |Wi,j| ›››› 2 ψα À log2pnpq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Let Jn “ max1ďiďn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1ďj,ℓďp ζ´2 1 |Wi,jWi,ℓ ´ EpWi,jWi,ℓq|, and σ2 n “ max 1ďj,ℓďp ζ´2 1 nÿ i“1 EtWi,jWi,ℓ ´ EpWi,jWi,ℓqu2 À max 1ďj,ℓďp ζ´2 1 nÿ i“1 Et|Wi,jWi,ℓ|2u À n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It also follows that }Jn}ψα{2 À ζ´2 1 ›››› max 1ďiďn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1ďj,ℓďp |Wij,Wi,ℓ| ›››› ψα{2 ` max 1ďiďn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1ďj,ℓďp ζ´2 1 Ep|Wi,jWi,ℓ|q À log2pnpq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' By Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 in Chernozhukov, Chetverikov and Kato (2017), it holds that Ep∆n1q À n´1 ” σn log1{2ppq ` tEpJ2 nqu1{2 log p ı À n´1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' n1{2 log1{2ppq ` log1{α`1pnpq ) À n´1{2 log1{2pnpq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then applying Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 in Chernozhukov, Chetverikov and Kato (2017) with η “ 1 and 34 β “ α{2, we obtain that Pp∆n1 ě 2Ep∆nq ` tq À exp ` ´C1nt2˘ ` 3 exp !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' ´C2ttn log´2{αpnpquα{2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, there exist a constant C1 depends on δ such that P !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' ∆n1 ą C1n´1{2 log1{2pnpq ) À p´δ Ñ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' From the multiplier bootstrap theorem and Gaussian comparison in Chernozhukov, Chetverikov and Kato (2017) and Koike (2021), sup tPRp ˇˇˇˇˇP˚ # n´1{2ζ´1 1 nÿ i“1 ZipWi ´ ¯Wnq ď t + ´ PpG ď tq ˇˇˇˇˇ À ∆1{2 n logppq ` tn´1 log5pnpqu1{4 , on t∆n À n´1{2 log1{2pnpqu, which occurs with probability 1 ´ p´δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Finally, similar to the proof of Theorem 1, we can show that under Conditions C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3 with a0ppq — p1´δ, if log p “ opn1{5q and log n “ opp1{3^δq, we have sup APAre ˇˇˇPtn1{2pˆθn ´ θq P Au ´ P˚pn1{2˜θn P Aq ˇˇˇ Ñ 0 in probability, which completes the proof of this theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Theorems 1 and 2 indicates that there exists a positive sequence βn,p Ñ 0 as n, p Ñ 8 such that sup tPR ˇˇˇPpn1{2|ˆθn ´ θ|8 ď tq ´ Pp|G|8 ď tq ˇˇˇ ď βn,p{2 and sup tPR ˇˇˇPpn1{2|ˆθn ´ θ|8 ď tq ´ P˚pn1{2|˜θn|8 ď tq ˇˇˇ ď βn,p with probability approaching one when n Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Letting q1´α be the p1 ´ αqth quantile of 35 n1{2|ˆθn ´ θ|8, that is, q1´α “ inftu P R : Ppn1{2|ˆθn ´ θ|8 ą uq ď αu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then, P˚pn1{2|˜θn|8 ď q1´α`βn,pq ě Ppn1{2|ˆθn ´ θ|8 ď q1´α`βn,pq ´ βn,p ě 1 ´ α , with probability approaching one as n Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' it holds with the same probability that P˚pn1{2|˜θn|8 ď q1´α´3βn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='pq ď Ppn1{2|ˆθn ´ θ|8 ď q1´α´3βn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='pq ` βn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='p “ Ppn1{2|ˆθn ´ θ|8 ď q1´α´3βn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='p ´ n´1{6q ` βn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='p `Ppn1{2|ˆθn ´ θ|8 ď q1´α´3βn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='pq ´Ppn1{2|ˆθn ´ θ|8 ď q1´α´3βn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='p ´ n´1{6q ă 1 ´ α ´ 2βn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='p ` Ppn1{2|ˆθn ´ θ|8 ď q1´α´3βn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='pq ´Ppn1{2|ˆθn ´ θ|8 ď q1´α´3βn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='p ´ n´1{6q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' where Ppn1{2|ˆθn ´ θ|8 ď q1´α´3βn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='pq ´ Ppn1{2|ˆθn ´ θ|8 ď q1´α´3βn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='p ´ n´1{6q can be bounded by ˇˇˇPpn1{2|ˆθn ´ θ|8 ď q1´α´3βn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='pq ´Ppn1{2|ˆθn ´ θ|8 ď q1´α´3βn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='p ´ n´1{6q ˇˇˇ ď ˇˇˇPp|G|8 ď q1´α´3βn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='pq ´ Pp|G|8 ď q1´α´3βn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='p ´ n´1{6q ˇˇˇ ` ˇˇˇPpn1{2|ˆθn ´ θ|8 ď q1´α´3βn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='pq ´ Pp|G|8 ď q1´α´3βn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='pq ˇˇˇ ` ˇˇˇPpn1{2|ˆθn ´ θ|8 ď q1´α´3βn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='p ´ n´1{6q ´ Pp|G|8 ď q1´α´3βn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='p ´ n´1{6q ˇˇˇ ď ˇˇˇPp|G|8 ď q1´α´3βn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='pq ´ Pp|G|8 ď q1´α´3βn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='p ´ n´1{6q ˇˇˇ ` βn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='p ď C1 ␣ n´1 log5pnpq (1{6 ` βn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' for some positive constant C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' where the last inequality follows from the Nazarov’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 36 Choosing C1 ␣ n´1 log5pnpq (1{6 ď βn,p, we obtain P˚pn1{2|˜θn|8 ď q1´α´3βn,pq ă 1 ´ α with probability approaching one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It follows that Ppq1´α´3βn,p ă qB 1´α ď q1´α`βn,pq Ñ 1, as n, p Ñ 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Therefore, Ppn1{2|ˆθn ´ θ|8 ą qB 1´αq ď Ppn1{2|ˆθn ´ θ|8 ą q1´α´3βn,pq ` PpqB 1´α ď q1´α´3βn,pq ď α ` 3βn,p ` op1q (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='14) and Ppn1{2|ˆθn ´ θ|8 ą qB 1´αq ě Ppn1{2|ˆθn ´ θ|8 ą q1´α`βn,pq ´ PpqB 1´α ą q1´α`βn,pq ě Ppn1{2|ˆθn ´ θ|8 ą q1´α`βn,p ´ n´1{6q ´ op1q `Ppn1{2|ˆθn ´ θ|8 ą q1´α`βn,pq ´Ppn1{2|ˆθn ´ θ|8 ą q1´α`βn,p ´ n´1{6q ě α ´ 2βn,p ´ C2 ␣ n´1 log5pnpq (1{6 ě α ´ 3βn,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' for some positive constant C2, where the second last inequality follows from the Nazarov’s inequality and the last inequality is from choosing βn,p ě C2 ␣ n´1 log5pnpq (1{6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Finally, as βn,p Ñ 0, Ppn1{2|ˆθn ´ θ|8 ě qB 1´αq ´ α Ñ 0, which completes the proof of this theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Without loss of generality, we assume θ0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Rewrite the test statistic as Tn “ n1{2|ˆθn|8, and let T c n “ n1{2|ˆθn ´ θ|8, which has the same distribution of Tn under H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 37 Then, it holds that Tn ě n1{2|θ|8 ´ T c n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Therefore, the power of the test based on Tn satisfies PpTn ą qB 1´α | H1q ě Ppn1{2|θ|8 ´ T c n ě qB 1´α | H1q “ PpT c n ď n1{2|θ|8 ´ qB 1´α | H1q Under the conditions of Theorem 2, there exists a positive sequence βn,p Ñ 0 as n, p Ñ 8, satisfies sup tPR |PpT c n ą t | H1q ´ Pp|G|8 ą t | H1q| ď βn,p, (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='15) where G „ Np0, ζ´2 1 Bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Letting q1´α be the p1 ´ αqth quantile of T C n and qG 1´α be the p1 ´ αqth quantile of |G|8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Choosing t “ qG 1´α`2βn,p in equation (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='15), we obtain that |PpT c n ą qG 1´α`2βn,p | H1q ´ α ` 2βn,p| ď βn,p and PpT c n ą qG 1´α`2βn,p | H1q ď α ´ βn,p, which implies that q1´α`βn,p ď qG 1´α`2βn,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Note that qB 1´α is the p1´αqth quantile of n1{2|˜θn|8 conditional on X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' By carrying out similar procedure as in the proof of equation (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='14), we can show that PpT c n ą n1{2|θ|8 ´ qB 1´α | H1q ď PpT c n ą n1{2|θ|8 ´ q1´α`βn,p | H1q ` op1q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='16) It follows that PpT c n ą n1{2|θ|8 ´ qB 1´α | H1q ď PpT c n ą n1{2|θ|8 ´ qG 1´α`2βn,p | H1q ` op1q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' For |G|8, we know that }|G|8}ψ2 À log1{2pnpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In addition, for any t ą 0, Pp|G|8 ą tq ď 2 expt´C1pt{}|G|8}ψ2q2u ď 2 expt´C2t2 log´1pnpqu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Choosing t “ C´1{2 2 log1{2p2{pα ´ 2βn,pqq log1{2pnpq, we arrive at Pp|G|8 ą C´1{2 2 log1{2p2{pα ´ 2βn,pqq log1{2pnpqq ď α ´ 2βn,p , 38 which leads to qG 1´α`2βn,p ď C´1{2 2 log1{2p2{pα ´ 2βn,pqq log1{2pnpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then, if |θ|8 ě Cn´1{2 log1{2pα´1q log1{2pnpq for a large enough constant C, it holds with sufficiently large C3 that PpTn ą qB 1´α | H1q ě PpT c n ď n1{2|θ|8 ´ qG 1´α`2βn,p | H1q ` op1q ě PtT c n ď C3 log1{2pnpq log1{2pα´1q | H1u ` op1q ě Pt|G|8 ď C3 log1{2pnpq log1{2pα´1q | H1u ´ βn,p ` op1q ě 1 ´ 2αC2C2 3 ´ βn,p ` op1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' We complete the proof of this theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Recall that ˆζ1 “ n´1 řn i“1 }Xi ´ ˆθn}´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It has been shown in the proof of Lemma A5 that }Xi ´ ˆθn}´1 “ R´1 i ´ 1 ` R´1 i W J i ˆθn ´ 2´1R´2 i }ˆθn}2 ` ˜δ1i ¯ , where ˜δ1i satisfies ˜δ1i “ Oppn´1q and max1ďiďn ˜δ1i “ Oppn´1{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, ˆζ1 “ n´1 nÿ i“1 R´1 i ´ 1 ` R´1 i W J i ˆθn ´ 2´1R´2 i }ˆθn}2 ` ˜δ1i ¯ “ n´1 nÿ i“1 R´1 i p1 ` ˜δ3iq, where ˜δ3i satisfies ˜δ3i “ Oppn´1{2q and max1ďiďn ˜δ3i “ Oppn´1{4q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' By the fact that n´1 řn i“1 R´1 i “ ζ1 ` Oppζ1n´1{2q, we conclude that ˆζ1{ζ1 ´ 1 “ Oppn´1{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 39 Let ˜Wi “ pXi ´ ˆθnq{}Xi ´ ˆθn} for i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' From the proof of Lemma A5, ˜Wi “ pWi ´ R´1 i ˆθnqp1 ` ˜δ2iq “ Wi ` Wi˜δ2i ´ R´1 i ˆθnp1 ` ˜δ2iq, where ˜δ2i satisfies ˜δ2i “ Oppn´1{2q and max1ďiďn ˜δ2i “ Oppn´1{4q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Let ˜Wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='j be the jth component of ˜Wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' then ˆBjj “ n´1 nÿ i“1 ˜W 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='j “ n´1 nÿ i“1 W 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='jt1 ` Opp˜δ2iqu ` n´1 nÿ i“1 R´1 i Wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='j ˆθn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='jt1 ` Opp˜δ2iqu `n´1 nÿ i“1 R´2 i ˆθ2 n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='jt1 ` Opp˜δ2iqu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' where max1ďjďp n´1 řn i“1 W 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='jt1 ` Opp˜δ2iqu{Bjj “ 1 ` Oppn´1{4q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' max 1ďjďp ˇˇˇˇˇn´1 nÿ i“1 R´2 i ˆθ2 n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='jt1 ` Opp˜δ2iqu ››››› À ˇˇˇˇˇn´1 nÿ i“1 R´2 i ˇˇˇˇˇ max 1ďjďp |ˆθ2 n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='j| “ Optζ2 1n´1 log1{2pnpqu and max 1ďjďp ˇˇˇˇˇn´1 nÿ i“1 R´1 i Wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='j ˆθn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='jt1 ` Opp˜δ2iqu ˇˇˇˇˇ À ˜ n´1 nÿ i“1 R´2 i ¸1{2 max 1ďjďp $ & % ˜ n´1 nÿ i“1 W 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='j ¸1{2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' max 1ďjďp |ˆθn,j| “ Optζ1p´1{2n´1{2 log1{2pnpqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It follows that max 1ďjďp ˆBjj{Bjj “ 1 ` Optn´1{4 log1{2pnpqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 40 Thus, Condition A (ii) of Belloni et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='al (2018) is satisfied by sn,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It is clear that Condition A (i) of Belloni et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='al (2018) is satisfied by the remainder term Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Hence, from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 in Belloni et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='al (2018), for any 1 ď j ď p, if log p “ opn1{5q and log n “ opp1{3^δq, we have sup 0ďxď21{2 log1{2pnpq ˇˇˇP !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' n1{2pˆθn,j ´ θjq{sn,j ą x ) ´ t1 ´ Φpxqu ˇˇˇ Ñ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Let ¯Tj “ n´1{2 řn i“1 Wi,j{tn´1 řn i“1 W 2 i,j ´ pn´1 řn i“1 Wi,jq2u1{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Based on Equation (13) of Liu and Shao (2014), for any sequence dn Ñ 8 and dn “ oppq as n Ñ 8, with Condition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4, sup 0ďtďG´1 κ pdn{pq ˇˇˇˇˇ ř jPH0 It| ¯Tj| ě tu p0Gκptq ´ 1 ˇˇˇˇˇ “ opp1q , where Gκptq is some function such that Gκptq ě Gptq “ 2t1 ´ Φptqu for all t P R, and Gκptq “ Gptqt1 ` op1qu uniformly over 0 ď t ď 21{2 log1{2ppq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then, with enough large n, as long as |H| Ñ 8 and |H| ą 2{α, we have α|H|{p ě 2{p “ 2 expt´p21{2 log1{2 pq2{2u ě 2t1 ´ Φp21{2 log1{2 pqu “ Gp21{2 log1{2 pq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It follows that G´1pα|H|{pq ď 21{2 log1{2 p, and consequently, sup 0ďtďG´1pα|H|{pq ˇˇˇˇ ř jPH0 It| ¯Tj| ě tu p0Gptq ´ 1 ˇˇˇˇ “ opp1q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Let T 1 j “ n1{2pˆθn,j ´ θjq{sn,j, we obtain max1ďjďp |T 1 j ´ ¯Tj| “ optlog´1{2ppqu with some careful calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' With similar procedure to Page 84 of Belloni et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='al (2018), it holds that sup 0ďtďG´1pα|H|{pq ˇˇˇˇ ř jPH0 It|T 1 j| ě tu p0Gptq ´ 1 ˇˇˇˇ “ opp1q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='17) The B-H method with P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , Pp is equivalent to the following procedure: reject H0j, if only if Pj ď ˆt0, where ˆt0 “ sup " 0 ď t ď 1 : t ď α maxtřp j“1 IpPj ď tqu p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then we have ˆt0 “ α maxtřp j“1 IpPjďˆt0q,1u p , and α|H|{p ě Gp21{2 log1{2 pq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Set t “ G´1pα|H|{pq, 41 then t ď 21{2 log1{2 p with probability tends to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, we have Gptq “ α|H| p ď α maxtřp j“1 Ip|Ti| ě 21{2 log1{2 pq, 1u p ď α maxtřp j“1 Ip|Ti| ě tq, 1u p , where the second inequality implied by (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='29) of Belloni et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='al (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It implies that Ppˆt0 ě α|H|{pq Ñ 1 with ˆt0 “ Gpˆtq, and together with (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='17), we have ř jPH0 It|T 1 j| ě ˆtu p0Gpˆtq “ ř jPH0 Ip|Tj| ě ˆtq p0Gpˆtq Ñ 1 , which is equivalent to ř jPH0 IpPj ď ˆt0q p0ˆt0 Ñ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Finally, FDRM “ ř jPH0 IpPj ď ˆt0q maxtřp j“1 IpPj ď ˆt0q, 1u “ ř jPH0 IpPj ď ˆt0q pˆt0{α Ñ αp0 p as n Ñ 8, which completes the proof of this theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Appendix B: Proof of preliminary lemmas In this section, we present proofs of preliminary lemmas in Section A1 of Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Proof of Lemma A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' As the components of U1 are independent and standardized, simple cal- culations yield Ep}U1}2q “ p and Ep}ΓU1}2q “ EpU J 1 ΓJΓU1q “ trtΓJΓEpU1U J 1 qu “ trpΩq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Under Condition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1, the components of U1 “ pU1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , U1,pqJ are independent sub-exponential random variables such that max1ďjďp }U1,j}ψα ď c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Applying the concentration inequality in the proof of Lemma S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 in (Wang, Peng and Li 2015), for every t ě 0, P `ˇˇ}U1}2 ´ p ˇˇ ě t ˘ ď C1 exp !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' ´C2 ` p´1t2˘α{p4α`4q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='18) 42 and P ␣ˇˇ}ΓU1}2 ´ trpΩq ˇˇ ě t ( ď C1 exp „ ´C2 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' t2 trpΩ2q )α{p4α`4qȷ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='19) For any fixed 0 ă ϵ ă 1, let A1 “ tp ´ ϵpp1`δq{2 ď }U1}2 ď p ` ϵpp1`δq{2u and A2 “ tp1 ´ ϵqtrpΩq ď }ΓU1}2 ď p1 ` ϵqtrpΩqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Taking t “ ϵpp1`δq{2 in (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='18) and t “ ϵtrpΩq in (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='19), we have PpA1q ě 1 ´ C1 exp !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' ´C2pϵ2pδqα{p4α`4q) and PpA2q ě 1 ´ C1 exp « ´C2 "ϵ2tr2pΩq trpΩ2q α{p4α`4qff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Under Condition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3, trpΩ2q “ pÿ j“1 pÿ ℓ“1 ω2 jℓ ď ¯ Mp max 1ďℓďp pÿ j“1 |ωjℓ| ď ¯ Mpa0ppq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Since trpΩq “ p and a0ppq — p1´δ, we conclude that tr2pΩq trpΩ2q ě p2 ¯ Mpa0ppq — pδ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Consequently, for some positive constants c1 and c2, we get that PpA1q ě 1 ´ c1 expt´c2pδα{p4α`4qu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 43 and PpA2q ě 1 ´ c1 expt´c2pδα{p4α`4qu for sufficient large p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, we finish the proof of this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Proof of Lemma A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (i) As the components of Ui “ pUi,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , Ui,pqJ are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' standardized sub- exponential random variables, simple algebra yields Ep}Ui}4q “ E $ & % ˜ pÿ j“1 U 2 i,j ¸2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' “ pÿ j“1 EpU 4 i,jq ` ÿ 1ďj1‰j2ďp EpU 2 i,j1qEpU 2 i,j2q “ pEpU 4 i,jq ` ppp ´ 1q and Ep}Ui}6q “ pÿ j“1 EpU 6 i,jq ` 3 ÿ 1ďj1‰j2ďp EpU 4 i,j1qEpU 2 i,j2q ` ÿ 1ďj1‰j2‰j3ďp EpU 2 i,j1qEpU 2 i,j2qEpU 2 i,j3q “ pEpU 6 i,jq ` 3ppp ´ 1qEpU 4 i,jq ` ppp ´ 1qpp ´ 2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In addition, Ep}Ui}8q “ pÿ j“1 EpU 8 i,jq ` 4 ÿ 1ďj1‰j2ďp EpU 6 i,j1qEpU 2 i,j2q `3 ÿ 1ďj1‰j2ďp EpU 4 i,j1qEpU 4 i,j2q `6 ÿ 1ďj1‰j2‰j3ďp EpU 4 i,j1qEpU 2 i,j2qEpU 2 i,j3q ` ÿ 1ďj1‰j2‰j3‰j4ďp EpU 2 i,j1qEpU 2 i,j2qEpU 2 i,j3qEpU 2 i,j4q “ pEpU 8 i,jq ` 4ppp ´ 1qEpU 6 i,j1q ` 3ppp ´ 1qtEpU 4 i,j1qu2 `3ppp ´ 1qEpU 4 i,jq ` ppp ´ 1qpp ´ 2qpp ´ 3q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 44 The result of Ep}Ui}2kq “ pk ` Oppk´1q for any positive integer k can be checked by Ep}Ui}2kq “ ÿ 1ďj1‰¨¨¨‰jkďp EpU 2 i,j1q ˆ ¨ ¨ ¨ ˆ EpU 2 i,jkqt1 ` Opp´1qu “ pk ` Oppk´1q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Moreover, by the fact that t1 ` u ´ pu ´ 1q2u{2 ď u1{2 ď p1 ` uq{2 for all u ě 0, we can get that Ep}U}kq “ pk{2 ` Oppk{2´1q for all positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (ii) Write Λjℓ “ řp j1“1 Γj1jΓj1ℓ as the pj, ℓqth element of ΓJΓ, then Ep}ΓUi}4q “ E $ & % ˜ pÿ j“1 pÿ ℓ“1 ΛjℓUi,jUi,ℓ ¸2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' “ pÿ j“1 Λ2 jjEpU 4 i,jq ` 2 ÿ 1ďj1‰j2ďp Λ2 j1j2EpU 2 i,j1qEpU 2 i,j2q ` ÿ 1ďj1‰j2ďp Λj1j1Λj2j2EpU 2 i,j1qEpU 2 i,j2q “ EpU 4 i,jq pÿ j“1 Λ2 jj ` 2 ÿ 1ďj1‰j2ďp Λ2 j1j2 ` ÿ 1ďj1‰j2ďp Λj1j1Λj2j2 “ ˜ pÿ j“1 Λjj ¸2 ` tEpU 4 i,jq ´ 1u pÿ j“1 Λ2 jj ` 2 ÿ 1ďj1‰j2ďp Λ2 j1j2 “ ttrpΩqu2 ` OttrpΩ2qu as řp j“1 Λ2 jj`ř 1ďj1‰j2ďp Λ2 j1j2 “ řp j“1 řp ℓ“1 Λ2 jℓ “ trpΩ2q and trpΩ2q À p2´δ based on Condition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Similarly, we can show that Ep}ΓUi}6q “ ÿ 1ďj1‰j2‰j3ďp pΛj1j1Λj2j2Λj3j3 ` Λ2 j1j2Λj3j3 ` Λj1j2Λj1j3Λj2j3qEpU 2 i,j1qEpU 2 i,j2qEpU 2 i,j3qt1 ` Opp´1qu “ p3 ` Opp3´δq and Ep}ΓUi}12q “ p6 ` Opp6´δq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Similar to the proof of part (i), the result Ep}ΓUi}q “ p1{2`Opp1{2´δq and Ep}ΓUi}3q “ p3{2` Opp3{2´δq are directly consequences of Ep}ΓUi}2q “ p, Ep}ΓUi}4q “ p2 ` Opp2´δq, Ep}ΓUi}6q “ 45 p3 ` Opp3´δq, Ep}ΓUi}12q “ p6 ` Opp6´δq and t1 ` u ´ pu ´ 1q2u{2 ď u1{2 ď p1 ` uq{2 for all u ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (iii) Now we consider Et}ΓSpUiq}2u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' For i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , n, let A1i “ tp ´ ϵpp1`δq{2 ď }Ui}2 ď p ` ϵpp1`δq{2u for a fixed 0 ă ϵ ă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' According to Lemma A1 and the fact that }ΓUi}2 ď trpΩq}Ui}2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Et}ΓSpUiq}2u “ Ep}ΓUi}2}Ui}´2q “ p´1Et}ΓUi}2u ` E ␣ }ΓUi}2 ` }Ui}´2 ´ p´1˘( “ 1 ` E ␣ }ΓUi}2 ` }Ui}´2 ´ p´1˘( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' where E ␣ }ΓUi}2 ` }Ui}´2 ´ p´1˘( ď p´1E ` }ΓUi}2}Ui}´2 ˇˇ}Ui}2 ´ p ˇˇ˘ “ p´1E ␣ }ΓUi}2}Ui}´2 ˇˇ}Ui}2 ´ p ˇˇ IpA1iq ( `p´1E ␣ }ΓUi}2}Ui}´2 ˇˇ}Ui}2 ´ p ˇˇ IpAc 1iq ( ď p´1tp ´ ϵpp1`δq{2u´1E ` }ΓUi}2 ˇˇ}Ui}2 ´ p ˇˇ˘ `p´1trpΩqE ␣ˇˇ}Ui}2 ´ p ˇˇ IpAc 1iq ( ď p´1tp ´ ϵpp1`δq{2u´1 ␣ Ep}ΓUi}4q (1{2 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Ep ˇˇ}Ui}2 ´ p ˇˇ2q )1{2 ` !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Ep ˇˇ}Ui}2 ´ p ˇˇ2q )1{2 tPpAc 1iqu1{2 ď p´1pp ´ ϵp1´δq´1tp2 ` Opp2´δqu1{2 ˆ Opp1{2q `Opp1{2q ˆ c1{2 1 expt´c2pδα{p4α`4q{2u “ Opp´1{2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It follows that Et}ΓSpUiq}2u “ 1 ` Opp´1{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 46 Similarly, the last result follows from Et}ΓSpUiq}4u “ p´2Et}ΓUi}4u ` E ␣ }ΓUi}4 ` }Ui}´4 ´ p´2˘( “ 1 ` Opp´δq ` E ␣ }ΓUi}4 ` }Ui}´4 ´ p´2˘( , where E ␣ }ΓUi}4 ` }Ui}´4 ´ p´2˘( ď p´2E ` }ΓUi}4}Ui}´4 ˇˇ}Ui}4 ´ p2ˇˇ˘ “ p´2E ␣ }ΓUi}4}Ui}´4 ˇˇ}Ui}4 ´ p2ˇˇ IpA1iq ( `p´2E ␣ }ΓUi}4}Ui}´4 ˇˇ}Ui}4 ´ p2ˇˇ IpAc 1iq ( ď p´2pp ´ ϵp1´δq´2E ` }ΓUi}4 ˇˇ}Ui}4 ´ p2ˇˇ˘ `p´2ttrpΩqu2E ␣ˇˇ}Ui}4 ´ p2ˇˇ IpAc 1iq ( ď p´2pp ´ ϵp1´δq´2 ␣ Ep}ΓUi}6q (2{3 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Ep ˇˇ}Ui}4 ´ p2ˇˇ3q )1{3 ` !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Ep ˇˇ}Ui}4 ´ p2ˇˇ2q )1{2 tPpAc 1iqu1{2 ď p´2pp ´ ϵp1´δq´2 ˆ Opp2q ˆ Opp3{2q `Opp3{2q ˆ c1{2 1 expt´c2pδα{p4α`4q{2u “ Opp´1{3q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (iv) as νi and SpUiq are independent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Epν´1 i qEt}ΓSpUiq}´1u “ Epν´1 i }ΓUi}´1}Ui}q “ EpR´1 i }Ui}q “ EtR´1 i }Ui}IpA1iqu ` EtR´1 i }Ui}IpAc 1iqu ď tp ` ϵpp1`δq{2u1{2EtR´1 i IpA1iqu ` tEpR´4 i qu1{4tE}Ui}4u1{4tPpAc 1iqu1{2 À tp ` ϵpp1`δq{2u1{2EpR´1 i q ` ζ1{4 4 ˆ p1{2 ˆ c1{2 1 expt´c2pδα{p4α`4q{2u À ζ1p1{2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 47 and Epν´2 i qEt}ΓSpUiq}´2u “ EtR´2 i }Ui}2IpA1iqu ` EtR´2 i }Ui}2IpAc 1iqu ď tp ` ϵpp1`δq{2uEtR´2 i IpA1iqu ` tEpR´4 i qu1{2tE}Ui}6u1{3tPpAc 1iqu1{6 À tp ` ϵpp1`δq{2uEpR´2 i q ` ζ1{2 4 ˆ p ˆ c1{6 1 expt´c2pδα{p4α`4q{6u À ζ2p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' we also have Epν´3 i qEt}ΓSpUiq}´3u “ EtR´3 i }Ui}3IpA1iqu ` EtR´3 i }Ui}3IpAc 1iqu ď tp ` ϵpp1`δq{2u3{2EtR´3 i IpA1iqu ` tEpR´4 i qu3{4tE}Ui}18u1{6tPpAc 1iqu1{12 À tp ` ϵpp1`δq{2u3{2EpR´3 i q ` ζ3{4 4 ˆ p3{2 ˆ c1{12 1 expt´c2pδα{p4α`4q{12u À ζ3p3{2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' By Cauchy-Schwarz inequality and Jensen’s inequality, we can show that rEt}ΓSpUiq}´1us´1 ď Et}ΓSpUiq}u ď rEt}ΓSpUiq}2us1{2 “ 1 ` Opp´1{2q , rEt}ΓSpUiq}´2us´1 ď Et}ΓSpUiq}2u “ 1 ` Opp´1{2q , and rEt}ΓSpUiq}´3us´1 ď Et}ΓSpUiq}3u ď rEt}ΓSpUiq}4us3{4 “ 1 ` Opp´1{3q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then, the results of this part follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' We finish the proof of this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Proof of Lemma A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (i) For i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , n, let A2i “ tp1 ´ ϵqtrpΩq ď }ΓUi}2 ď p1 ` ϵqtrpΩqu for a fixed 0 ă ϵ ă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Recall that Γj is the jth row of Γ and Wi,j “ ΓjUi{}ΓUi}, then Qjℓ “ n´1 nÿ i“1 R´1 i Wi,jWi,ℓ “ n´1 nÿ i“1 ν´1 i pΓjUiqpΓℓUiq}ΓUi}´3 48 “ n´1p´3{2 nÿ i“1 ν´1 i pΓjUiqpΓℓUiq `n´1 nÿ i“1 ν´1 i pΓjUiqpΓℓUiq ´ }ΓUi}´3 ´ p´2{3¯ , where the last term satisfies ˇˇˇˇˇE # n´1 nÿ i“1 ν´1 i pΓjUiqpΓℓUiq ´ }ΓUi}´3 ´ p´2{3¯+ˇˇˇˇˇ ď p´3{2E !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' ν´1 i |pΓjUiqpΓℓUiq| }ΓUi}´3 ˇˇˇ}ΓUi}3 ´ p3{2ˇˇˇ ) “ p´3{2E !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' R´1 i |pΓjUiqpΓℓUiq| }ΓUi}´2 ˇˇˇ}ΓUi}3 ´ p3{2ˇˇˇ ) “ p´3{2E !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' R´1 i |pΓjUiqpΓℓUiq| }ΓUi}´2 ˇˇˇ}ΓUi}3 ´ p3{2ˇˇˇ IpA2iq ) `p´3{2E !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' R´1 i |pΓjUiqpΓℓUiq| }ΓUi}´2 ˇˇˇ}ΓUi}3 ´ p3{2ˇˇˇ IpAc 2iq ) ď p1 ´ ϵq´1p´5{2E !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' R´1 i |pΓjUiqpΓℓUiq| ˇˇˇ}ΓUi}3 ´ p3{2ˇˇˇ IpA2iq ) `p´3{2E !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' R´1 i ˇˇˇ}ΓUi}3 ´ p3{2ˇˇˇ IpAc 2iq ) À p´5{2tEpR´4 i qu1{4 ” E !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' |pΓjUiqpΓℓUiq|4)ı1{4 " E ˆˇˇˇ}ΓUi}3 ´ p3{2ˇˇˇ 2˙*1{2 `p´3{2tEpR´4 i qu1{4 " E ˆˇˇˇ}ΓUi}3 ´ p3{2ˇˇˇ 2˙*1{2 tPpAc 2iqu1{4 À ζ1p´1´δ{2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It follows that Qjℓ “ n´1p´3{2 nÿ i“1 ν´1 i pΓjUiqpΓℓUiq ` Oppζ1p´1´δ{2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' For i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , n, let A1i “ tp ´ ϵpp1`δq{2 ď }Ui}2 ď p ` ϵpp1`δq{2u for a fixed 0 ă ϵ ă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' According to Lemma A1, E ”␣ ΓjSpUiqSpUiqJΓJ ℓ (2ı “ E ␣ }Ui}´4pΓjUiU J i ΓJ ℓ q2( “ E ␣ }Ui}´4pΓjUiU J i ΓJ ℓ q2IpA1iq ( ` E ␣ }Ui}´4pΓjUiU J i ΓJ ℓ q2IpAc 1iq ( À tp ´ ϵpp1`δq{2u´2E ␣ pΓjUiU J i ΓJ ℓ q2( ` p2PpAc 1iq 49 À tp ´ ϵpp1`δq{2u´2 ` p2 ˆ c1 expt´c2pδα{p4α`4qu À p´2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then, we can show that n´1p´3{2 nÿ i“1 ν´1 i pΓjUiqpΓℓUiq “ n´1p´1{2 nÿ i“1 ν´1 i ΓjSpUiqSpUiqJΓJ ℓ ` Oppζ1p´7{6q , where the last equality is indicated by E|p´3{2ν´1 i ΓjSpUiqSpUiqJΓJ ℓ p}Ui}2 ´ pq| À p´3{2tEpν´3 i qu1{3 ´ E ”␣ ΓjSpUiqSpUiqJΓJ ℓ (2ı¯1{2 “ E ␣ p}Ui}2 ´ pq6(‰1{6 À ζ1p´7{6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, we obtain that Qjℓ “ n´1p´1{2 nÿ i“1 ν´1 i ΓjSpUiqSpUiqJΓJ ℓ ` Oppζ1p´7{6 ` ζ1p´1´δ{2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' As νi and SpUiq are independent with each other, we have E # n´1p´1{2 nÿ i“1 ν´1 i ΓjSpUiqSpUiqJΓJ ℓ + “ p´1{2Epν´1 i qE ␣ ΓjSpUiqSpUiqJΓJ ℓ ( , where Epν´1 i q À p1{2ζ1 from Lemma A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' According to Lemma A1 and regarding that ΓjΓJ ℓ “ ωjℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='␣ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ΓjSpUiqSpUiqJΓJ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='` ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ΓjUiU J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='i ΓJ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ℓ }Ui}´2˘ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='p´1E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='` ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ΓjUiU J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='i ΓJ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='˘ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='` E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='␣ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ΓjUiU J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='i ΓJ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='` ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='}Ui}´2 ´ p´1˘( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='p´1ωjℓ ` E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='␣ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ΓjUiU J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='i ΓJ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='` ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='}Ui}´2 ´ p´1˘( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ď ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='p´1|ωjℓ| ` E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='`ˇˇΓjUiU J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='i ΓJ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ˇˇ ˇˇ}Ui}´2 ´ p´1ˇˇ˘ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='p´1|ωjℓ| ` p´1E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='`ˇˇΓjUiU J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='i ΓJ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ˇˇ }Ui}´2 ˇˇ}Ui}2 ´ p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ˇˇ˘ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='p´1|ωjℓ| ` p´1E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='␣ˇˇΓjUiU J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='i ΓJ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ˇˇ }Ui}´2 ˇˇ}Ui}2 ´ p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ˇˇ IpA1iq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='`p´1E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='␣ˇˇΓjUiU J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='i ΓJ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ˇˇ }Ui}´2 ˇˇ}Ui}2 ´ p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ˇˇ IpAc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1iq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='À ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='p´1|ωjℓ| ` tp2 ´ ϵpp3`δq{2u´1E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='`ˇˇΓjUiU J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='i ΓJ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ˇˇ ˇˇ}Ui}2 ´ p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ˇˇ˘ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='`E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='␣ˇˇ}Ui}2 ´ p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ˇˇ IpAc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1iq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ď ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='p´1|ωjℓ| ` tp2 ´ ϵpp3`δq{2u´1 ” ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='` ΓjUiU J i ΓJ ℓ ˘2)ı1{2 ” E !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='` }Ui}2 ´ p ˘2)ı1{2 ` ” E !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='` }Ui}2 ´ p ˘2)ı1{2 tPpAc 1iqu1{2 ď p´1|ωjℓ| ` Opp´3{2q ` Opp1{2q ˆ c1{2 1 expt´c2pδα{p4α`4q{2u À p´1|ωjℓ| ` Opp´3{2q , where the second last inequality is due to E !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='` }Ui}2 ´ p ˘2) “ Ep}Ui}4 ´ 2p}Ui}2 ` p2q “ pEpU 4 i,jq ` ppp ´ 1q ´ 2p2 ` p2 “ Oppq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, it follows that E # n´1p´1{2 nÿ i“1 ν´1 i ΓjSpUiqSpUiqJΓJ ℓ + À ζ1p´1|ωjj| ` Opζ1p´3{2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Furthermore, as Epν´2 i q À pζ2, we can conclude that Var # n´1p´1{2 nÿ i“1 ν´1 i ΓjSpUiqSpUiqJΓJ ℓ + “ n´1p´1Epν´2 i qE ”␣ ΓjSpUiqSpUiqJΓJ ℓ (2ı ´n´1p´1tEpν´1 i qu2 “ E ␣ ΓjSpUiqSpUiqJΓJ ℓ (‰2 À ζ2 1n´1p´2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 51 It follows from the Chebychev’s inequality that ˇˇˇˇˇn´1p´1{2 nÿ i“1 ν´1 i ΓjSpUiqSpUiqJΓJ ℓ ˇˇˇˇˇ À ζ1p´1|ωjℓ| ` Oppζ1n´1{2p´1 ` ζ1p´3{2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Finally, we arrive at |Qj,ℓ| À ζ1p´1|ωjℓ| ` Oppζ1n´1{2p´1 ` ζ1p´7{6 ` ζ1p´1´δ{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (ii) From the proof of part (i), we know that Qjℓ “ Q0,jℓ ` Oppζ1p´7{6 ` ζ1p´1´δ{2q , where Q0,jℓ is the pj, ℓqth component of the random matrix Q0 “ n´1p´1{2 řn i“1 ν´1 i tΓSpUiqutΓSpUiquJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In addition, E ␣ ΓjSpUiqSpUiqJΓJ ℓ ( À p´1|ωjℓ| ` Opp´3{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It follows that tr !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='` E “ tΓSpUiqutΓSpUiquJ‰˘2) “ pÿ j“1 pÿ ℓ“1 “ E ␣ ΓjSpUiqSpUiqJΓJ ℓ (‰2 À p´2 pÿ j“1 pÿ ℓ“1 |ωjℓ|2 ` p´5{2 pÿ j“1 pÿ ℓ“1 |ωjℓ| ` p´1 À p´1a0ppq ` p´3{2a0ppq ` p´1 À p´δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' This implies that trrtEpQ0qu2s “ p´1tEpν´1 i qu2tr !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='` E “ tΓSpUiqutΓSpUiquJ‰˘2) À p´1´δ and EttrpQ2 0qu “ n´1p´1tr ` E “ ν´2 i tΓSpUiqutΓSpUiquJtΓSpUiqutΓSpUiquJ‰˘ `p1 ´ n´1qp´1tr !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='` E “ ν´1 i tΓSpUiqutΓSpUiquJ‰˘2) “ n´1p´1Epν´2 i qE ␣ }ΓSpUiq}4( `p1 ´ n´1qp´1tEpν´1 i qu2tr !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='` E “ tΓSpUiqutΓSpUiquJ‰˘2) “ Opn´1p´1q ` trrtEpQ0qu2sp1 ´ n´1q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 52 Thus, we have trrEpQ2 0q ´ tEpQ0qu2s “ Opn´1p´1q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' We complete the proof of this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Proof of Lemma A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Recall that Γj is the jth row of Γ, and denote Γjℓ to be the pj, ℓqth element of Γ, then ΓjUi “ pÿ ℓ“1 ΓjℓUi,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It is noticed that ωjℓ “ řp j1“1 Γjj1Γℓj1, then VarpΓjUiq “ pÿ ℓ“1 Γ2 jℓ “ ωjj and EtpΓjUiq4u “ E $ & % ˜ pÿ ℓ“1 ΓjℓUi,ℓ ¸4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' “ pÿ ℓ“1 Γ4 jℓEpU 4 i,ℓq ` 6 ÿ 1ďℓ1‰ℓ2ďp Γ2 jℓ1Γ2 jℓ2EpU 2 i,ℓ1qEpU 2 i,ℓ2q À ω2 jj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (i) For i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , n, let A2i “ tp1 ´ ϵqtrpΩq ď }ΓUi}2 ď p1 ` ϵqtrpΩqu for a fixed 0 ă ϵ ă 1, then PpA2iq ě 1 ´ c1 expt´c2pδα{p4α`4qu according to the proof of Lemma A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It follows that EpW 4 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='jq “ Et}ΓUi}´4pΓjUiq4u “ Et}ΓUi}´4pΓjUiq4IpA2iqu ` Et}ΓUi}´4pΓjUiq4IpAc 2iqu 53 ď tp1 ´ ϵqtrpΩqu´2EtpΓjUiq4u ` PpAc 2iq À ω2 jjtp1 ´ ϵqtrpΩqu´2 ` c1 expt´c2pδα{p4α`4qu À ω2 jjttrpΩqu´2 and EpW 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='jq ě Et}ΓUi}´2pΓjUiq2IpA2iqu ě tp1 ` ϵqtrpΩqu´1EtpΓjUiq2IpA2iqu “ tp1 ` ϵqtrpΩqu´1EtpΓjUiq2u ´ tp1 ` ϵqtrpΩqu´1EtpΓjUiq2IpAc 2iqu ě tp1 ` ϵqtrpΩqu´1EtpΓjUiq2u ´ tp1 ` ϵqtrpΩqu´1rEtpΓjUiq4us1{2tPpAc 2iqu1{2 Á ωjjtp1 ` ϵqtrpΩqu´1 ´ tp1 ` ϵqtrpΩqu´1 ˆ ωjj ˆ c1{2 1 expt´c2pδα{p4α`4q{2u Á ωjjttrpΩqu´1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' from which we conclude that Etpζ´1 1 Wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='jq4u À ζ´4 1 p´2ω2 jj À ¯ M2 and Etpζ´1 1 Wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='jq2u Á ζ´2 1 p´1ωjj Á m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (ii) Similar to the proof of part (i), for any ϱ ě 1, E ␣ |ζ´1 1 Wi,j|ϱ( “ E ␣ |ζ´1 1 Wi,j|ϱIpA1iq ( ` E ␣ |ζ´1 1 Wi,j|ϱIpAc 1iq ( À ζ´ϱ 1 ttrpΩqu´ϱ{2Et|ΓjUi|ϱu ` ζ´ϱ 1 PpAc 1iq À Et|ΓjUi|ϱu ` pϱ{2 expt´c2pδα{p4α`4qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Since max1ďjďp }Ui,j}ψα ď c0 for some constant c0, we have }ΓjUi}ψα À c0 according to Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 in Koike (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then, we known that Et|ΓjUi|ϱu À ϱϱ{α for any ϱ ě 1 by the equivalent 54 sub-exponential properties (Koike 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Therefore, E ␣ |ζ´1 1 Wi,j|ϱ( À ϱϱ{α for any ϱ ě 1 for sufficient large p, which indicates that ζ´1 1 Wi,j is sub-exponential, and thus }ζ´1 1 Wi,j}ψα À ¯B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (iii) By simple algebra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' EpW 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='jq “ p´1EtpΓjUiq2u ` EtpΓjUiq2p}ΓUi}´2 ´ p´1qu “ p´1ωjj ` EtpΓjUiq2p}ΓUi}´2 ´ p´1qu ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' where EtpΓjUiq2p}ΓUi}´2 ´ p´1qu satisfies ˇˇEtpΓjUiq2p}ΓUi}´2 ´ p´1qu ˇˇ ď p´1EtpΓjUiq2}ΓUi}´2|}ΓjUi}2 ´ p|u “ p´1EtpΓjUiq2}ΓUi}´2|}ΓjUi}2 ´ p|IpA2iqu `p´1EtpΓjUiq2}ΓUi}´2|}ΓjUi}2 ´ p|IpAc 2iqu ď p´1tp1 ´ ϵqtrpΩqu´1EtpΓjUiq2|}ΓjUi}2 ´ p|u ` p´1Et}ΓjUi}2 ´ p|IpAc 2iqu ď p´2p1 ´ ϵq´1rEtpΓjUiq4us1{2tEp|}ΓjUi}2 ´ p|2qu1{2 `p´1tEp|}ΓjUi}2 ´ p|2qu1{2tPpAc 2iqu1{2 À p´2 ˆ p1´δ{2 ` p´1 ˆ p1´δ{2 ˆ c1{2 1 expt´c2pδα{p4α`4q{2u À p´1´δ{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' for 1 ď j ‰ ℓ ď p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' we have EpWi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='jWi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ℓq “ p´1EtpΓjUiqpΓℓUiqu ` EtpΓjUiqpΓℓUiqp}ΓUi}´2 ´ p´1qu “ p´1ωjℓ ` EtpΓjUiqpΓℓUiqp}ΓUi}´2 ´ p´1qu ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='55 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='where EtpΓjUiqpΓℓUiqp}ΓUi}´2 ´ p´1qu satisfies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ˇˇEtpΓjUiqpΓℓUiqp}ΓUi}´2 ´ p´1qu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ˇˇ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ď ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='p´1Et|pΓjUiqpΓℓUiq|}ΓUi}´2|}ΓjUi}2 ´ p|u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='p´1Et|pΓjUiqpΓℓUiq|}ΓUi}´2|}ΓjUi}2 ´ p|IpA2iqu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='`p´1Et|pΓjUiqpΓℓUiq|}ΓUi}´2|}ΓjUi}2 ´ p|IpAc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2iqu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ď ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='p´1tp1 ´ ϵqtrpΩqu´1Et|pΓjUiqpΓℓUiq||}ΓjUi}2 ´ p|u ` p´1Et}ΓjUi}2 ´ p|IpAc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2iqu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='ď ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='p´2p1 ´ ϵq´1rEt|pΓjUiqpΓℓUiq|2us1{2tEp|}ΓjUi}2 ´ p|2qu1{2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='`p´1tEp|}ΓjUi}2 ´ p|2qu1{2tPpAc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2iqu1{2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='À ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='p´2 ˆ p1´δ{2 ` p´1 ˆ p1´δ{2 ˆ c1{2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='expt´c2pδα{p4α`4q{2u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='À ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='p´1´δ{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (iv) According to part (ii), ζ´1 1 W1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , ζ´1 1 Wn are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' p-dimensional random vectors satis- fies }ζ´1 1 Wi,j}ψα À ¯B for all i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , n and j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 of van der Vaart & Wellner (1996), ›››› max 1ďiďn max 1ďjďp |ζ´1 1 Wi,j| ›››› ψα À log1{αpnpq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Similar to the proof of part (i), we can show that Etpζ´1 1 Wi,jq2u “ ζ´2 1 Et}ΓUi}´2pΓjUiq2IpA1iqu `ζ´2 1 Et}ΓUi}´4pΓjUiq4IpAc 1iqu ď ζ´2 1 tp1 ` ϵqtrpΩqu´1EtpΓjUiq2u ` ζ´2 1 EtIpAc 1iqu ď ζ´2 1 ωjjtp1 ` ϵqtrpΩqu´1 ` ζ´2 1 c1 expt´c2pδ{p4`4αqu “ ζ´2 1 ωjjtp1 ` ϵqtrpΩqu´1t1 ` op1qu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It follows that max 1ďjďp nÿ i“1 Etpζ´1 1 Wi,jq2u À max 1ďjďp nÿ i“1 ζ´2 1 ωjjtp1 ` ϵqtrpΩqu´1 À n max 1ďjďp ωjj ď ¯ Mn , 56 Applying Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 of Chernozhukov, Chetverikov and Kato (2017), it holds that with α ě 1 and n´1{2 log3{2pnpq À 1, E ˜ˇˇˇˇˇn´1{2 nÿ i“1 ζ´1 1 Wi ˇˇˇˇˇ 8 ¸ À n´1{2tn1{2 log1{2ppq ` log1{αpnpq logppqu À log1{2pnpq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' From the properties of the ψα norm, it holds that ›››› max 1ďiďn,1ďjďp |ζ´1 1 Wi,j|2 ›››› ψα{2 À log2pnpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' According to Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3 of Chernozhukov, Chetverikov and Kato (2017), we have that E ˜ˇˇˇˇˇn´1 nÿ i“1 pζ´1 1 Wiq2 ˇˇˇˇˇ 8 ¸ À n´1t ¯ Mn ` log2pnpq logppqu À ¯ M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' We finish the proof of this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Proof of Lemma A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Let ˜Xi “ Xi ´ ˆθn and ˜Ri “ } ˜Xi} for i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' According to the proof of Lemma 1, }ˆθn} “ Oppζ´1 1 n´1{2q and max1ďiďn R´1 i “ Oppζ1n1{4q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then R´1 i }ˆθn} satisfies R´1 i }ˆθn} “ Oppn´1{2q and max 1ďiďn R´1 i }ˆθn} “ Oppn´1{4q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' As ˜R´1 i “ R´1 i }Wi ´R´1 i ˆθn}´1 “ R´1 i ´ 1 ´ 2R´1 i W J i ˆθn ` R´2 i }ˆθn}2¯´1{2 , by Taylor expansion, ˜R´1 i “ R´1 i ´ 1 ` R´1 i W J i ˆθn ´ 2´1R´2 i }ˆθn}2 ` ˜δ1i ¯ , where ˜δ1i satisfies ˜δ1i “ Oppn´1q and max1ďiďn ˜δ1i “ Oppn´1{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It follows that ˜R´1 i “ R´1 i p1 ` ˜δ2iq , where ˜δ2i “ R´1 i W J i ˆθn ´ 2´1R´2 i }ˆθn}2 ` ˜δ1i satisfies ˜δ2i “ Oppn´1{2q and max1ďiďn ˜δ2i “ 57 Oppn´1{4q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, ˜R´1 i “ Oppζ1q and max 1ďiďn ˜R´1 i “ Oppζ1n1{4q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Denote ˜Wi “ ˜Xi{} ˜Xi} for i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then, ˜Wi “ ˜R´1 i pXi ´ ˆθnq “ R´1 i pXi ´ ˆθnqp1 ` ˜δ2iq “ pWi ´ R´1 i ˆθnqp1 ` ˜δ2iq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' We first show that }˜θn} “ Oppζ´1 1 n´1{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It is noticed that ˜θn minimizes L˚ npβq “ nÿ i“1 }Zi ˜Xi ´ β} , which is a strictly convex function of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, if we can show that L˚ npβq has a ζ1n1{2-consistent local minimizer, then this local minimizer must be a ζ1n1{2-consistent global minimizer of L˚ npβq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The existence of a ζ1n1{2-consistent local minimizer is implied by the fact that for an arbitrarily small ε ą 0, there exists a constant C0, which does not depend on n and p, such that lim inf n P " inf qPRp, }q}“C0 L˚ npζ´1 1 n´1{2qq ą L˚ np0q ą 1 ´ ε, (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='20) Since |Zi| “ 1, we rewrite }Zi ˜Xi ´ ζ´1 1 n´1{2q} as }Zi ˜Xi ´ ζ´1 1 n´1{2q} “ ˜Ri ´ 1 ´ 2ζ´1 1 n´1{2 ˜R´1 i ZiqJ ˜Wi ` ζ´2 1 n´1 ˜R´2 i }q}2¯1{2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' As |ζ´1 1 n´1{2 ˜R´1 i ZiqT ˜Wi| “ Oppn´1{2q and ζ´2 1 n´1 ˜R´2 1i }q}2 “ Oppn´1q, by Taylor expansion, we obtain that }Zi ˜Xi ´ ζ´1 1 n´1{2q} “ ˜Ri ´ ζ´1 1 n´1{2ZiqJ ˜Wi ` 2´1ζ´2 1 n´1 ˜R´1 i }q}2 58 ´2´1ζ´2 1 n´1 ˜R´1 i qJ ˜Wi ˜W J i q ` Oppζ´1 1 n´3{2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then, ζ1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' L˚ npζ´1 1 n´1{2qq ´ L˚ np0q ) “ ζ1 nÿ i“1 ´ }Zi ˜Xi ´ ζ´1 1 n´1{2q} ´ } ˜Xi} ¯ “ ´n´1{2qJ ˜ nÿ i“1 Zi ˜Wi ¸ ` 2´1ζ´1 1 n´1}q}2 nÿ i“1 ˜R´1 i ´2´1ζ´1 1 n´1qJ ˜ nÿ i“1 ˜Ri ˜Wi ˜W J i ¸ q ` Oppn´1{2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='21) As E˚ ´ n´1{2 řn i“1 Zi ˜Wi ¯ “ 0 and E˚ ¨ ˝ ›››››n´1{2 nÿ i“1 Zi ˜Wi ››››› 2˛ ‚“ n´1 nÿ i“1 ˜W J i ˜Wi “ 1, we obtain that ˇˇˇˇˇn´1{2qJ nÿ i“1 Zi ˜Wi ˇˇˇˇˇ ď }q} ›››››n´1{2 nÿ i“1 Zi ˜Wi ››››› “ Opp}q}q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In the meanwhile, as ζ´1 1 n´1 řn i“1 R´1 i “ 1 ` Oppn´1{2q, we have ζ´1 1 n´1}q}2 nÿ i“1 ˜R´1 i “ ζ´1 1 n´1}q}2 nÿ i“1 R´1 i p1 ` ˜δ2iq “ }q}2t1 ` Oppn´1{4qu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Simple algebra yields n´1 nÿ i“1 ˜R´1 i ˜Wi ˜W J i “ n´1 nÿ i“1 R´1 i pWi ´ R´1 i ˆθnqpWi ´ R´1 i ˆθnqJp1 ` ˜δ2iq 59 “ n´1 nÿ i“1 RiWiW J i p1 ` ˜δ2iq ´ 2n´1 nÿ i“1 R´2 i Wiˆθ J np1 ` ˜δ2iq `n´1 ÿ i“1 R´3 i ˆθnˆθ J np1 ` ˜δ2iq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Similar to the proof in Cheng et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='al (2019) and utilizing the results on Q “ n´1 řn i“1 R´1 i WiW ´1 i in Lemma A3, we can show that n´1qJ řn i“1 RiWiW J i qp1 ` ˜δ2iq “ Oppζ1n´1{2 ` ζ1p´p1{6^δ{2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In addition, as n´1 nÿ i“1 R´2 i qJWi ď n´1 nÿ i“1 R´2 i }q}}Wi} “ }q}n´1 nÿ i“1 R´2 i “ Oppζ2 1q and n´1 řn i“1 R´3 i “ ζ3t1 ` opp1qu, we have n´1qJ nÿ i“1 R´2 i Wiˆθ J nqp1 ` ˜δ2iq “ n´1 nÿ i“1 R´2 i qJWip1 ` ˜δ2iqpˆθ J nqq “ Oppζ1n´1{2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' and n´1qJ ÿ i“1 R´3 i ˆθnˆθ J nqp1 ` ˜δ2iq “ n´1 ÿ i“1 R´3 i p1 ` ˜δ2iq}qJˆθn}2 “ Oppζ1n´1q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, we obtain 2´1ζ´1 1 n´1}q}2 nÿ i“1 ˜R´1 i ` 2´1ζ´1 1 n´1qJ ˜ nÿ i“1 ˜Ri ˜Wi ˜W J i ¸ q “ 2´1}q}2 ` Oppn´1{4 ` p´δq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Choosing a sufficient large constant C0, the second term dominates the first term in (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='21) and thus ζ1 ␣ L˚ npζ´1 1 n´1{2qq ´ L˚ np0q ( ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Hence, we have }˜θn} “ Oppζ´1 1 n´1{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Denote Θi “ Ziˆθn ` ˜θn for i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then max 1ďiďn }Θi} ď }ˆθn} ` }˜θn} “ Oppζ´1 1 n´1{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 60 Recall that ˜θn satisfies nÿ i“1 Zi ˜Xi ´ ˜θn }Zi ˜Xi ´ ˜θn} “ nÿ i“1 ZiWi ´ R´1 i Θi }ZiWi ´ R´1 i Θi} “ 0 , which is equivalently to n´1 nÿ i“1 pZiWi ´ R´1 i Θiq ` 1 ´ 2ZiR´1 i W J i Θi ` R´2 i }Θi}2˘´1{2 “ 0 , where |R´1 i W J i Θi| “ Oppn´1{2q, R´2 i }Θi}2 “ Oppn´1q, max 1ďiďn |R´1 i W J i Θi| “ Oppn´1{4q and max 1ďiďn R´2 i }Θi}2 “ Oppn´1{2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Taylor expansion leads to n´1 nÿ i“1 pZiWi ´ R´1 i Θiqp1 ` ZiR´1 i W J i Θi ´ 2R´2 i }Θi}2 ` ˜δ3iq “ 0 where δ3i “ OptpZiR´1 i W J i Θi ´ R´2 i }Θi}2q2u “ Oppn´1q, and max1ďiďn δ3i “ Oppn´1{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Then, n´1 nÿ i“1 ZiWip1 ´ 2R´2 i }Θi}2 ` ˜δ3iq ` n´1 nÿ i“1 R´1 i pW J i ΘiqWi “ n´1 nÿ i“1 ZiWip1 ´ 2R´2 i }Θi}2 ` ˜δ3iq ` n´1 nÿ i“1 ZiR´1 i WiW J i ˆθn `n´1 nÿ i“1 R´1 i WiW J i ˜θn “ n´1 nÿ i“1 R´1 i Θip1 ` ˜δ3i ` ˜δ4iq “ n´1 nÿ i“1 R´1 i ˜θnp1 ` ˜δ3i ` ˜δ4iq ` n´1 nÿ i“1 ZiR´1 i ˆθnp1 ` ˜δ3i ` ˜δ4iq , where ˜δ4i “ ZiR´1 i W J i Θi ´ 2R´2 i }Θi}2 “ Opp˜δ1{2 3i q satisfies max1ďiďn ˜δ4i “ Oppn´1{4q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The proof of Lemma 1 implies |ˆθ|8 “ Optn´1{2 log1{2pnpqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' As E˚ ` n´1 řn i“1 ZiR´1 i ˘ “ 0 and E˚ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='` n´1 řn i“1 ZiR´1 i ˘2) “ n´2 řn i“1 R´2 i “ Oppn´1ζ2q, we have n´1 řn i“1 ZiR´1 i “ Oppζ1n´1{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 61 As Zi is bounded, it is straightforward to show that |n´1{2 řn i“1 ZiWi|8 “ Optp´1{2 log1{2pnpqu similar as in the proof of Lemma A4 (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Thus, similar to the proof of Lemma 1, we obtain that |˜θ|8 “ Optn´1{2 log1{2pnpqu and ˇˇˇˇˇn´1 nÿ i“1 R´1 i WiW J i ˜θn ˇˇˇˇˇ 8 “ Optζ1n´1{2p´p1{6^δ{2q log1{2pnpq ` ζ1n´1 log1{2pnpqu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In the meanwhile, it holds that |n´1 řn i“1 R´1 i | “ ζ1 ` Oppζ1n´1{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Finally, n1{2˜θn “ n´1{2ζ´1 1 nÿ i“1 ZiWi ` ˜Cn , (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='22) and ˜Cn is the remainder term satisfies | ˜Cn|8 “ Optn´1{4 log1{2pnpq ` p´δ´p1{6^δ{2q log1{2pnpqu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' We finish the proof of this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Appendix C: Additional simulation results In this section, we report additional simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Section C1 presents simulation results on SCIs for ρ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Section C2 reports simulations on global tests for high-dimensional location parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 Addition simulation results on simultaneous confidence intervals Tables A4 reports the coverage probability and median length of the SCIs based on ˆθn for ρ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5, the results of the SCIs based on the sample mean ¯Xn are presented in parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' We observe that the performance of the SCIs based on ˆθn with ρ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 is similar to 62 that of ρ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 in the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The SCIs achieve satisfactory coverage probability, and it is much shorter than those based on ¯Xn under the multivariate t-distribution, which is heavy-tailed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Table A4: Coverage probability (in %) and median length of the SCIs based on ˆθn, the results of the SCIs based on ¯Xn are in parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' θ “ θ1 θ “ θ2 Coverage probability Median length Coverage probability Median length Model ρ n p 90% 95% 90% 95% 90% 95% 90% 95% I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 100 100 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='9) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='65 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='65) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='69 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='69) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='65 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='65) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='69 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='69) 1000 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='77 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='77) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='80 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='80) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='77 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='77) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='80 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='80) 200 100 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='9) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='46 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='46) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='49 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='49) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='46 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='46) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='49 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='49) 1000 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='55 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='55) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='57 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='57) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='55 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='55) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='57 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='57) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 100 100 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='65 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='65) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='69 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='69) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='65 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='65) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='69 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='69) 1000 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='77 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='77) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='80 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='80) 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='77 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='77) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='80 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='80) 200 100 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='9 (90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='9) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='46 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='46) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='49 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='49) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7 (90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3 (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='46 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='46) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='49 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='49) 1000 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='55 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='55) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='57 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='57) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='55 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='55) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='57 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='57) II 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 100 100 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='71 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='05) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='12) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='71 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='05) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='11) 1000 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='84 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='24) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='88 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='30) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='9) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='84 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='24) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='88 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='30) 200 100 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='76) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='53 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='80) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='76) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='53 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='80) 1000 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='59 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='90) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='62 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='93) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='59 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='90) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='62 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='94) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 100 100 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 (87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='9) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 (93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='9) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='71 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='05) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='12) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='71 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='05) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='11) 1000 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='9) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='84 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='24) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='88 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='30) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='84 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='88 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='30) 200 100 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='76) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='53 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='80) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='76) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='53 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='80) 1000 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='9) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='59 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='90) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='62 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='94) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='59 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='89) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='62 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='93) III 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 100 100 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='65 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='66) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='69 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='70) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='65 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='66) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='69 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='70) 1000 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='78 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='78) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='82 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='82) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3 (90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='9) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='78 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='78) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='82 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='82) 200 100 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='46 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='46) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='49 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='49) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='46 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='46) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='49 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='49) 1000 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='55 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='55) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='57 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='58) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='55 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='55) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='57 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='57) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 100 100 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='9 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='65 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='65) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='69 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='69) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='65 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='65) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='69 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='69) 1000 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='78 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='78) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='81 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='81) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='9) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='78 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='78) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='81 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='81) 200 100 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='46 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='46) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='49 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='49) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='7) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='9 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='46 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='46) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='49 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='49) 1000 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='55 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='55) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='57 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='57) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='3 (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='6) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='4 (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='55 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='55) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='57 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='57) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2 Simulations on global tests for high-dimensional location parameters In this section, we report the performance of the test based on Tn (Median test) for one-sample high-dimensional location parameters, and compare it with three alternative approaches: the test of Chen and Qin (2010, CQ test);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' the test based on TMean (Mean test) and bootstrap approximation for ¯Xn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' the test of Wang, Peng and Li (2015, WPL test) based on TWPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' We consider the same data generation models (I, II and III) as in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' For θ, we set its first tc0 log pu components as non-zero, while the other elements are all zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' c0 is chosen from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The magnitude of non-zero entries in θ is κplog p{nq1{2, where κ is chosen from 0 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Note that κ “ 0 refers to the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' We consider n “ 50 or 100, and p “ 100 and 1000 for each sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 63 Figures A3–A10 plot the empirical size (κ “ 0) and power (κ ‰ 0) of four (CQ, Mean, Median, and WPL) tests at the 5% significance level for Models I and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The results of κ “ 0 indicates that the empirical sizes of all these four tests are close to the nominal significance level under different case scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' When κ ‰ 0, the power of these tests increases as κ increases, that is, as the signal getting stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' For Gaussian data, the Mean test based on TMean and the Median test based on Tn have similar power performances, and they advance both the CQ test and the WPL test, which are L2-norm type tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In addition, when the data are from multivariate t-distribution, the Median test outperforms the Mean test, which shows the superiority of the procedure based on the sample spatial median over that based on the sample mean under heavy-tailedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' In summary, the Median test based on Tn is preferred among the four tests when the alternative is sparse and the underlying distribution is heavy-tailed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Second, Figure A11 depicts empirical size and power of the four tests (CQ, Mean, Median, WPL) for Model III with ρ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' It can be seen that, even Model III is not a member of the elliptical distribution family, the size of the Median test can still control the size at the nominal level α “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='05, and this is also the case for the WPL test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' We can also see that the Median test and the Mean test have better power performance than the CQ test and the WPL test, especially for c0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 when the number of non-zero element in θ is relatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' References Belloni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Chernozhukov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Chetverikov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Hansen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', and Kato, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2018) High- dimensional econometrics and generalized GMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' arXiv preprint arXiv:1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='01888.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' and Qin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2010) A two-sample test for high-dimensional data with applications to gene-set testing, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Statist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 38 (2), 808–835.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Cheng,G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Liu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Peng, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Zhang, B and Zheng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='Testing the equality of two high- dimensional spatial sign covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Scand J Statist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 46, 257–271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Chernozhukov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Chetverikov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', and Kato, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Central limit theorems and bootstrap in high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The Annals of Probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 45(4), 2309–2352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 50 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 50 , p = 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 100 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 100 , p = 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 50 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 50 , p = 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 100 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 100 , p = 1000 Method CQ Mean Median WPL Figure A3: Empirical size and power of the four tests (CQ, Mean, Median, WPL) for Models I and II with c0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 and ρ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The horizontal black solid line refers to the nominal 5% significance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' “Gaussian” denotes the multivariate normal distribution, and t3 denotes the multivariate t-distribution with 3 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 50 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 50 , p = 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 100 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 100 , p = 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 50 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 50 , p = 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 100 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 100 , p = 1000 Method CQ Mean Median WPL Figure A4: Empirical size and power of the four tests (CQ, Mean, Median, WPL) for Models I and II with c0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 and ρ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The horizontal black solid line refers to the nominal 5% significance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' “Gaussian” denotes the multivariate normal distribution, and t3 denotes the multivariate t-distribution with 3 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 50 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 50 , p = 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 100 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 100 , p = 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 50 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 50 , p = 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 100 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 100 , p = 1000 Method CQ Mean Median WPL Figure A5: Empirical size and power of the four tests (CQ, Mean, Median, WPL) for Models I and II with c0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 and ρ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The horizontal black solid line refers to the nominal 5% significance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' “Gaussian” denotes the multivariate normal distribution, and t3 denotes the multivariate t-distribution with 3 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 50 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 50 , p = 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 100 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 100 , p = 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 50 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 50 , p = 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 100 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 100 , p = 1000 Method CQ Mean Median WPL Figure A6: Empirical size and power of the four tests (CQ, Mean, Median, WPL) for Models I and II with c0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 and ρ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The horizontal black solid line refers to the nominal 5% significance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' “Gaussian” denotes the multivariate normal distribution, and t3 denotes the multivariate t-distribution with 3 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 50 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 50 , p = 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 100 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 100 , p = 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 50 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 50 , p = 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 100 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 100 , p = 1000 Method CQ Mean Median WPL Figure A7: Empirical size and power of the four tests (CQ, Mean, Median, WPL) for Models I and II with c0 “ 1 and ρ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The horizontal black line refers to the nominal 5% significance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' “Gaussian” denotes the multivariate normal distribution, and t3 denotes the multivariate t-distribution with 3 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 50 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 50 , p = 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 100 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 100 , p = 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 50 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 50 , p = 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 100 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 100 , p = 1000 Method CQ Mean Median WPL Figure A8: Empirical size and power of the four tests (CQ, Mean, Median, WPL) for Models I and II with c0 “ 1 and ρ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The horizontal black solid line refers to the nominal 5% significance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' “Gaussian” denotes the multivariate normal distribution, and t3 denotes the multivariate t-distribution with 3 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 50 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 50 , p = 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 100 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 100 , p = 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 50 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 50 , p = 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 100 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 100 , p = 1000 Method CQ Mean Median WPL Figure A9: Empirical size and power of the four tests (CQ, Mean, Median, WPL) for Models I and II with c0 “ 1 and ρ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The horizontal black solid line refers to the nominal 5% significance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' “Gaussian” denotes the multivariate normal distribution, and t3 denotes the multivariate t-distribution with 3 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 50 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 50 , p = 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 100 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability Gaussian, n = 100 , p = 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 50 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 50 , p = 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 100 , p = 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability t3 , n = 100 , p = 1000 Method CQ Mean Median WPL Figure A10: Empirical size and power of the four tests (CQ, Mean, Median, WPL) for Models I and II with c0 “ 1 and ρ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The horizontal black line refers to the nominal 5% significance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' “Gaussian” denotes the multivariate normal distribution, and t3 denotes the multivariate t-distribution with 3 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability n = 50 , p = 100 , c0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability n = 50 , p = 100 , c0 = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability n = 50 , p = 1000 , c0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability n = 50 , p = 1000 , c0 = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability n = 100 , p = 100 , c0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability n = 100 , p = 100 , c0 = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability n = 100 , p = 1000 , c0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='00 0 1 2 3 4 5 κ Rejection Probability n = 100 , p = 1000 , c0 = 1 Method CQ Mean Median WPL Figure A11: Empirical size and power of the four tests (CQ, Mean, Median, WPL) for Model III with ρ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' The horizontal black solid line refers to the nominal 5% significance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 73 Koike, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Notes on the dimension dependence in high-dimensional central limit theorems for hyperrectangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Japanese Journal of Statistics and Data Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 1, 257–297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Liu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' and Shao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Phase transition and regularized bootstrap in large scale t-tests with false discovery rate control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Annals of Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 42, 2003–2025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Rudelson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', and Vershynin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2013) Hanson–Wright inequality and sub-Gaussian concen- tration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Electronic Communications in Probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 18, 1–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Vershynin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' High-Dimensional Probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Cambridge University Press, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=', Peng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' and Li, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' A high-dimensional nonparametric multivariate test for mean vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' Journal of the American Statistical Association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 110, 1658–1669.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} +page_content=' 74' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9E1T4oBgHgl3EQfXQT3/content/2301.03126v1.pdf'} diff --git a/H9AyT4oBgHgl3EQfrvmy/vector_store/index.faiss b/H9AyT4oBgHgl3EQfrvmy/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..f80fededc4a17333357fa425b8e13e532d27cc0e --- /dev/null +++ b/H9AyT4oBgHgl3EQfrvmy/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d396f3f539ed36b579b2e57bd79bc12ac8b6e5afcafef5038269154702710272 +size 5242925 diff --git a/H9FJT4oBgHgl3EQfFSyX/content/tmp_files/2301.11442v1.pdf.txt b/H9FJT4oBgHgl3EQfFSyX/content/tmp_files/2301.11442v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0db8d80182029310c5eaeb7f82f81a6e5f3ef7c3 --- /dev/null +++ b/H9FJT4oBgHgl3EQfFSyX/content/tmp_files/2301.11442v1.pdf.txt @@ -0,0 +1,1799 @@ +arXiv:2301.11442v1 [cs.LG] 26 Jan 2023 +Collaborative Regret Minimization in Multi-Armed Bandits +Nikolai Karpov 1 Qin Zhang 1 +Abstract +In this paper, we study the collaborative learning +model, which concerns the tradeoff between par- +allelism and communication overhead in multi- +agent reinforcement learning. For a fundamen- +tal problem in bandit theory, regret minimiza- +tion in multi-armed bandits, we present the first +and almost tight tradeoffs between the number of +rounds of communication between the agents and +the regret of the collaborative learning process. +1. Introduction +One of the biggest challenges with reinforcement learn- +ing is scalability. +In recent years, a series of papers +(Tao et al., 2019; Karpov et al., 2020; Wang et al., 2020; +Karpov & Zhang, 2022a;b) studied bandit problems in the +collaborative learning (CL) model, where multiple agents +interact with the environment to learn simultaneously and +cooperatively. One of the most expensive resources in the +CL model is communication, which consists of the number +of communication steps (round complexity) and the total +bits of messages exchanged between agents (bit complex- +ity). Communication directly contributes to learning time +(network bandwidth constraints and latency), energy con- +sumption (communication is frequently the biggest energy +drain for tasks such as deep-sea/outer-space exploration), +as well as data usage (if messages are sent by mobile de- +vices). In this paper, we focus on the round complexity in +the CL model and consider a basic problem in the bandit +theory named regret minimization in multi-armed bandits +(MAB for short). We give almost tight round-regret trade- +offs for MAB in the CL model. +The CL model is closely related to the batched learning +model, which has recently received considerable attention +in bandit theory and reinforcement learning (Perchet et al., +2015; Jun et al., 2016; Agarwal et al., 2017; Jin et al., 2019; +Gao et al., 2019; Esfandiari et al., 2019; Bai et al., 2019; +Karpov & Zhang, 2020; Jin et al., 2021). +The batched +model is motivated by applications in which there is a sig- +1Computer Science Department, Indiana University, Bloom- +ington, +USA. Correspondence to: +Nikolai Karpov , Qin Zhang . +nificant delay in getting back the observations, such as clin- +ical trials (Thompson, 1933; Robbins, 1952) and crowd- +sourcing (Kittur et al., 2008). We will illustrate the con- +nection between the two models shortly. +In the rest of this section, we will first introduce the CL +model and the MAB problem that we are going to study +in this paper. We then describe our results and place them +within the context of the literature. +Regret Minimization in MAB. +In the single-agent learn- +ing model, we have one agent and a set I of N arms; +the i-th arm is associated with a distribution Di with sup- +port [0, 1] and (unknown) mean µi. +At each time step +t = 1, 2, . . ., T , the agent pulls an arm πt and receives +an i.i.d. reward rt from distribution Dπt. Let [n] denote +{1, 2, . . ., n}. Let A be the algorithm the agent employs. +The goal is for the agent to minimize the expected regret +E[Reg(A(I, T ))] = E + + � +t∈[T ] +(µ⋆ − µπt) + + , +(1) +where µ⋆ ≜ max +i∈[N]{µi}. +Let E[Reg(A∗(I, T ))] be the +smallest expected regret that a single-agent algorithm can +achieve on input I under time horizon T . +The Collaborative Learning Model. +The CL model was +formalized in Tao et al. (2019). In this model, we have K +agents and N arms. The learning proceeds in R rounds for +a predetermined value R. Within each round, at each time +step, each agent k (k ∈ [K]) pulls an arm π(k) +t +based on its +previous pull outcomes and messages received from other +agents. At the end of each round, the K agents communi- +cate with each other to exchange newly observed informa- +tion and determine the number of time steps for the next +round; the number of time steps for the first round is fixed +at the beginning. Note that no communication is needed at +the end of the last round, and thus the number of communi- +cation steps is (R − 1). +The expected regret of a T -time K-agent collaborative al- + +Tight Bounds for Collaborative Regret Minimization Multi-Armed Bandits +gorithm AK on input I is defined to be +E [Reg(AK(I, T ))] = E + + � +t∈[T ] +� +k∈[K] +� +µ⋆ − µπ(k) +t +� + + . +(2) +Let E [Reg(A∗ +K(I, T ))] be the expected regret that the best +K-agent collaborative algorithm can achieve. Naturally, +we have +E [Reg(A∗ +K(I, T ))] ≤ K · E [Reg(A∗(I, T ))] , +since each agent can simply run the algorithm A without +communicating with other parties. On the other hand, since +any T -time K-agent collaborative algorithm can be simu- +lated by a (KT )-time single-agent algorithm, we have +E [Reg(A∗ +K, T )] ≥ E [Reg(A∗, KT )] . +The Batched Learning Model. +In the batched model, +there is one agent interacting with the arms. The learning +proceeds in batches. The sequence of arm pulls in each +batch need to be determined at the beginning of the batch. +The goal is for the agent to minimize the regret over a se- +quence of T pulls using a small number of batches. +The following observation connects the CL model and the +batched learning model. +Observation 1.1. If there is a T -time R-batch single-agent +algorithm that achieves an expected regret Reg(I) for any +input I, then there is a T +K -time R-round K-agent collab- +orative algorithm that achieves an expected regret Reg(I) +for any input I. +To see this, just note that each round of z (non-adaptive) +pulls in a batched algorithm can be evenly distributed to +the K agents in a collaborative algorithm so that each agent +makes z/K non-adaptive pulls. +Observation 1.1 allows us to establish a lower bound in the +batched model by proving a corresponding lower bound in +the CL model, and to design an algorithm for the CL model +using an algorithm for the batched model. +We note that Observation 1.1 is one-directional, and the +other direction does not hold. This is because the CL model +is stronger than the batched model in the sense that in +the CL model, each agent can make adaptive pulls in each +round. While in the batched model, the sequence of pulls +are non-adaptive in each batch. This key difference also +makes the previous approaches for proving lower bounds +for bandit problems in the batched model inapplicable to +the CL model. In the previous work (Tao et al., 2019), it +has been shown that for the problem of best arm identifica- +tion in multi-armed bandits, O(log K) rounds is sufficient +to achieve almost optimal instance-sensitive pull complex- +ity. However, Ω(log N) rounds is necessary to get almost +optimal instance-sensitive pull complexity in the batched +model. +Our Results. +Let ⋆ = arg maxa∈I µa. Let ∆a ≜ µ⋆ − +µa, and ∆(I) = mina̸=⋆ ∆a. All log’s have base 2 unless +specified explicitly. +In this paper, we give the following results: +1. Our main result is a lower bound for MAB in the CL +model (Theorem 3.1). We show that for any T -time +K-agent collaborative algorithm AK, there is an in- +put I such that if AK runs on I using at most R ∈ +� +4L +log K , +2L +log log L +� +rounds where L = +log(4KT ) +4 +, then +AK incurs an expected regret of ˜Ω +� +2 +4L +R · +1 +∆(I) +� +.1 +2. Using Observation 1.1, our lower bound for MAB in +the CL model also gives a lower bound for MAB in the +batched model (Corollary 3.2), which is comparable +to the previous best lower bound (Gao et al., 2019). +3. We propose an algorithm for batched MAB (Theo- +rem 4.1) with an instance-sensitive round complexity. +Again via Observation 1.1, we obtain an algorithm for +MAB in the CL model. Our upper bounds match the +lower bounds up to some logarithmic factors in regret. +2. Related Work +Work in the Collaborative Learning Model. +To the best +of our knowledge, the study of the CL model began from +Hillel et al. (2013), in which the authors considered the +problem of best arm identification (BAI) in MAB. How- +ever, Hillel et al. (2013) only considered the special case +that R = 2, and the CL model was formally defined in +this paper. The CL model that we use in this paper was in- +troduced by Tao et al. (2019), in which almost tight round- +time tradeoff was given for BAI in MAB.2 Karpov et al. +(2020) extended this line of work to the top-m arm iden- +tification in MAB. +Wang et al. (2020) studied regret minimization in MAB in +the CL model. However, their primary focus is the bit com- +plexity. The recent work Karpov & Zhang (2022a) investi- +gated the bit complexity of BAI in the CL model. +Several recent papers (Shi & Shen, 2021; Shi et al., 2021; +Karpov & Zhang, 2022b) studied problems in MAB in the +non-IID CL model, where agents interact with possibly dif- +ferent environments. More specifically, Shi & Shen (2021); +1We use ‘˜’ to hide non-critical logarithmic factors. All these +factors will be spelled out in the formal theorems in this paper. +2In Tao et al. (2019), the time cost was presented as speedup, +defined as the ratio between the running time of the collaborative +algorithm and that of the best centralized algorithm. + +Tight Bounds for Collaborative Regret Minimization Multi-Armed Bandits +Shi et al. (2021) studied regret minimization with a focus +on the bit complexity, but the bit cost in their model is +integrated into the regret formulation. +Karpov & Zhang +(2022b) gave almost tight round-time tradeoff for BAI in +the non-IID CL model. +Work in the Batched Learning Model. +Batched algo- +rithms for bandit problems have attracted significant atten- +tion in the past decade. As discussed previously, Gao et al. +(2019); Esfandiari et al. (2019) studied regret minimization +in MAB mentioned. An earlier work (Perchet et al., 2015) +studied the same problem on two arms. Jin et al. (2021) +considered asymptotic regret in MAB in the batched model. +Several recent papers studied batched regret minimization +in MAB using Thompson sampling (Kalkanli & ¨Ozg¨ur, +2021; Karbasi et al., 2021; Karpov & Zhang, 2021). An- +other series of works (Jun et al., 2016; Agarwal et al., 2017; +Jin et al., 2019) studied batched BAI in MAB. +Other Work in Multi-Agent Bandit Learning +There +are many other papers concerning multi-agent bandit learn- +ing, but they do not focus on the cost of communica- +tion between the agents. For example, R´eda et al. (2022) +gave collaborative algorithms for BAI and regret mini- +mization in MAB in the setting that we allow agents +to communicate at each time step. +A series of papers +(Sz¨or´enyi et al., 2013; Landgren et al., 2016; 2018) con- +sidered MAB problems in the peer-to-peer (P2P) comput- +ing models such that at each time step, agents can only +communicate with their neighbors in the P2P network. +Several papers (Liu & Zhao, 2010; Rosenski et al., 2016; +Bistritz & Leshem, 2018; Bubeck & Budzinski, 2020) con- +sidered the collision model, in which if multiple agents try +to pull the same arm at a particular time step, then their re- +wards will be reduced due to collision. A comprehensive +survey of multi-agent bandit learning is beyond the scope +of this paper. +3. The Lower Bound +In this section, we give the following theorem which is the +main result of this paper. +Theorem 3.1. For any T -time K-agent collaborative al- +gorithm AK, there is an input I such that if AK runs on +I using at most R rounds for any R ∈ +� +4L +log K , +2L +log log L +� +where L = log(4KT ) +4 +, then AK incurs an expected regret of +Ω +� +2 +4L +R +L log L · +1 +∆(I) +� +. +In Theorem 3.1, we have implicitly assumed that K ≥ +log log T . If K < log log T , then the regret lower bound +for (KT )-time single-agent algorithms for MAB, which is +also a lower bound for any T -time K-agent collaborative +algorithms, will match the regret of the trivial 1-round algo- +rithm for MAB in the CL model (i.e., each of the K agents +runs the best single-agent algorithm (Bubeck et al., 2013)) +up to a log log T factor in regret. +We note two special cases. For simplicity, we assume T ≥ +K, and thus log(KT ) = Θ(log T ). +• The “full communication” case. +When R += +2L +log log L = Θ +� +log T +log log log T +� +, AK incurs an expected +regret of Ω +� +log log T +log T +· +1 +∆(I) +� +. +An almost match- +ing upper bound (Theorem 4.1) indicates that this +lower bound is almost tight. These results show that +Θ +� +log T +log log log T +� +rounds is necessary and sufficient for +a collaborative algorithm to achieve a regret that is +comparable to the best single-agent algorithm that +runs in KT time (i.e., Reg(A∗, KT )). +• The “no communication” case. +When R += +4L +log K += +Θ +� +log T +log K +� +, AK incurs an expected re- +gret of Ω +� +1 +log T log log T · +K +∆(I) +� +. An almost tight up- +per bound can be achieved by letting each of the +K agents run the best single-agent algorithm for +MAB (Bubeck et al., 2013) independently without any +communication. These results indicate that to achieve +any non-trivial parallelism for MAB, the agents need +at least Ω +� +log T +log K +� +rounds of communication. +By Observation 1.1, we have the following corollary (set- +ting K = T +1 +2 in Theorem 3.1). +Corollary 3.2. For any T -time batched algorithm A for +MAB, there is an input I such that if A runs on I using at +most R ∈ +� +4, +2L +log log L +� +batches where L = +log(4T 1.5) +4 +, then +A incurs an expected regret of Ω +� +2 +4L +R +L log L · +1 +∆(I) +� +. +This result is comparable with the lower bound result for +(adaptive grid) batched algorithms in Gao et al. (2019). In +particular, both results show that Ω(log T/ log log T ) is +necessary to achieve the optimal regret O +� +log T · +1 +∆(I) +� +. +In the rest of this section, we try to prove Theorem 3.1. We +start by introducing some concepts and notations. +Transcript. +Let γ = ((j1, o1), . . . , (jn, on)) be a se- +quence of pulls and outcomes on some input I for MAB, +where jt is the index of the arm in I being pulled at the t-th +time step and ot is the corresponding reward. We call γ the +transcript of a sequence of arm pulls, and use |γ| = n to +denote the length of γ (i.e., the number of (jt, ot) pairs in +γ). For convenience, we use j(γ) = (j1, . . . , jn) to denote + +Tight Bounds for Collaborative Regret Minimization Multi-Armed Bandits +the sequence of arm indices and o(γ) to denote the corre- +sponding sequence of rewards. +For a sequence of arm indices j(γ), let ΘI(j(γ)) be the +sequence of (random) rewards by pulling the arms of I ac- +cording to j(γ). We define +gI(γ) ≜ Pr[ΘI(j(γ)) = o(γ)], +(3) +which is the probability of observing the reward sequence +o(γ) by pulling the arms in input I following the index se- +quence j(γ). +For a single-agent algorithm A for MAB, an input I and a +time horizon n, we use Γ ∼ A(I, n) to denote a random +transcript generated by running A on input I for n time +steps. For a K-agent collaborative algorithm AK, we write +Γ ∼ AK(I, n) as the round-robin concatenation of the K +transcripts generated by the K agents on input I for n time +steps. That is, +Γ = +� +(J(1) +1 , O(1) +1 ), . . . , (J(K) +1 +, O(K) +1 +), . . . , (J(K) +n +, O(K) +n +) +� +, +where (J(k) +t +, O(k) +t +) is the pull and outcome of agent k at t- +th time step; we use capital letters J(k) +t +and O(k) +t +since they +are random variables that depend on the previous pulls and +(random) outcomes. +Notations. +We will use R to denote the number of rounds +used by the K-agent collaborative algorithm AK. For a +time horizon T , we will create L = log4(4KT ) +2 +pairs of hard +inputs. We will consider R satisfying +4L +log K ≤ R ≤ +2L +log log L. +(4) +This inequality will be implicitly assumed in the rest of the +section. +We will use the following constants in the proof: ǫ = 10−1, +λ = 10−6, and β = 4. We will use the notations instead +of the actual constants in most places of this section for the +sake of readability. +3.1. The Hard Inputs +We start the proof by introducing the set of hard inputs. +For each ℓ ∈ {1, . . . , L} and each σ ∈ {+1, −1}, we de- +sign an input: +Iσ +ℓ = Ber +�1 +2 + σ +βℓ +� +⊗ Ber +�1 +2 − σ +βℓ +� +. +(5) +In words, Iσ +ℓ is an input on two Bernoulli arms (i.e., the +reward is either 0 or 1 on each pull), where the first arm has +mean 1 +2 + σ +βℓ and the second arm has mean 1 +2 − σ +βℓ . For the +convenience of writing, we will abbreviate I+1 +ℓ +and I−1 +ℓ +to +I+ +ℓ and I− +ℓ respectively. +For ℓ ∈ [L], let ∆ℓ = +2 +βℓ be the mean gap between the two +arms in the inputs I+ +ℓ (or I− +ℓ ). Thus, ℓ = logβ +1 +∆ℓ − 1. +We define the set of hard inputs to be +I = {I+ +1 , I− +1 , . . . , I+ +L , I− +L } . +We also use Iℓ = {I+ +ℓ , I− +ℓ , . . . , I+ +L , I− +L } to denote a suffix +of I starting from index ℓ. +The next two lemmas establish some properties of the in- +puts in I. +Both lemmas concern the quantity ln gA(γ) +gB(γ), +where A and B are two inputs in I and γ is a transcript. +The proofs of the two lemmas involve technical calculation. +Due to space constraints, we leave them to Appendix B. +Lemma 3.3. Fix any ℓ ∈ [L] and any pair of inputs +A, B ∈ Iℓ. For any transcript γn = γn−1 ◦ (jn, on) = +((j1, o1), . . . , (jn, on)) with gA(γn), gB(γn) > 0, it holds +that +����ln gA(γn) +gB(γn) − ln gA(γn−1) +gB(γn−1) +���� ≤ 5 +βℓ . +Let A be a single-agent algorithm for MAB and I be an +input, we use Γ ∼ A(I, n) to denote a random transcript Γ +generated by A which runs on input I for n time steps. +Lemma 3.4. Let A be any single-agent algorithm for MAB. +For any ℓ ∈ [L] and any inputs A, B, I ∈ Iℓ, consider the +random transcript Γn = Γn−1 ◦ (Jn, On) ∼ A(I, n), +E +Γn +� +ln gA(Γn) +gB(Γn) − ln gA(Γn−1) +gB(Γn−1) +���� Γn−1 +� +≤ 11 +β2ℓ . +3.2. Indistinguishable Input Pairs +We introduce the following event defined on a transcript γ. +Definition 3.5. Event E(γ): +For any ℓ ∈ [L] such that +λβ2ℓ +log L ≥ |γ|, and for any pair of inputs A, B ∈ Iℓ, we have +ln gA(γ) +gB(γ) ≤ 2ǫ . +Intuitively, it says that when the length of transcript γ is +smaller than λβ2ℓ +log L, the probabilities of producing γ under +all inputs in Iℓ are similar. We will often abbreviate E(γ) +to E when γ is clear from the context. +The following lemma states that for a random transcript Γ +generated by running a single-agent algorithm on any input +in I, E(Γ) holds with high probability. +Lemma 3.6. For any single-agent algorithm A for MAB, +any I ∈ I, and any n > 0, it holds that +Pr +Γ∼A(I,n)[E(Γ)] ≥ 1 − 1 +L6 . + +Tight Bounds for Collaborative Regret Minimization Multi-Armed Bandits +Proof. By a union bound, we have +Pr +Γ∼A(I,n)[¯E] ≤ +� +ℓ∈[L]: +λβ2ℓ +log L ≥n +� +A,B∈Iℓ +Pr +Γ∼A(I,n) +� +ln gA(Γ) +gB(Γ) > 2ǫ +� +. +(6) +We try to bound each term in the summation of (6). Con- +sider a fixed ℓ satisfying λβ2ℓ +log L ≥ n and a fixed pair of inputs +A, B ∈ Iℓ. Let Γn ∼ A(I, n), and Γt (t ∈ [n]) be the pre- +fix of Γn of length t. We introduce the following sequence +of random variables for t = 1, . . . , n: +Zt ≜ ln gA(Γt) +gB(Γt) − 11 +β2ℓ t . +We also define Z0 ≜ 0. +Claim 3.7. Z0, Z1, . . . , Zn form a supermartingale. +Proof. We write +E +Γt +[Zt − Zt−1 | Zt−1] += +E +Γt +[Zt − Zt−1 | Γt−1] += +E +Γt +� +ln gA(Γt) +gB(Γt) − ln gA(Γt−1) +gB(Γt−1) +���� Γt−1 +� +− 11 +β2ℓ +≤ +0, +where the last inequality follows from Lemma 3.4. +By Lemma 3.3, we have +|Zn − Zn−1| ≤ 5 +βℓ + 11 +β2ℓ ≤ 10 +βℓ . +Applying +Azuma’s +inequality +(Lemma +A.2) +on +Z0, Z1, . . . , Zn with d = 10/βℓ, we get +Pr[Zn − 0 ≥ ǫ] +≤ +exp +� +− +ǫ2 +2 · (10/βℓ)2 · n +� += +exp +� +−ǫ2β2ℓ +200n +� += +exp +� +−ǫ2 log L +200λ +� +≤ +1 +L10 . +Consequently, with probability 1 − 1/L10, we have +ln gA(Γt) +gB(Γt) +≤ +Zn + 11 +β2ℓ n +< +ǫ + ǫ = 2ǫ . +(7) +Combining (6) and (7), we have +Pr[¯E] ≤ L · (2L)2 · 1/L10 ≤ 1/L6. +The lemma follows. +The next lemma shows that short transcripts generated by a +single-agent algorithm on two inputs in Iℓ are statistically +indistinguishable. +Lemma 3.8. Let A be any single-agent algorithm for MAB. +For a transcript γ, let G(γ) be an event determined by γ. +For any ℓ ∈ [L], any pair of inputs A, B ∈ Iℓ, and any +n ≤ λβ2ℓ +log L, we have +Pr +Γ∼A(A,n)[G(Γ) ∧ E(Γ)] ≤ e2ǫ +Pr +Γ∼A(B,n)[G(Γ) ∧ E(Γ)] . +Proof. Let ΓA ∼ A(A, n) and ΓB ∼ A(B, n) be two ran- +dom transcripts. Based on our construction of hard inputs, +it is easy to see that supp(ΓA) = supp(ΓB). Define a set +of transcripts +W ≜ {γ | (γ ∈ supp(ΓA)) ∧ G(γ) ∧ E(γ)} . +By the law of total probability, we have +Pr +Γ∼A(A,n)[G(Γ) ∧ E(Γ)] = +� +γ∈W +Pr +Γ∼A(A,n)[Γ = γ], +and +Pr +Γ∼A(B,n)[G(Γ) ∧ E(Γ)] = +� +γ∈W +Pr +Γ∼A(B,n)[Γ = γ]. +Therefore, we only need to show +� +γ∈W +Pr +Γ∼A(A,n)[Γ = γ] ≤ e2ǫ � +γ∈W +Pr +Γ∼A(B,n)[Γ = γ]. (8) +By the definition of gI(γ) (Eq. (3)), we have +Pr +Γ∼A(A,n)[Γ = γ] = gA(γ) and +Pr +Γ∼A(B,n)[Γ = γ] = gB(γ). +(9) +When E(γ) holds, by Definition 3.5, +gA(γ) ≤ e2ǫgB(γ). +(10) +Inequality (8) follows from (9) and (10). +3.3. The Lower Bound Proof +Now we are ready to give the proof of Theorem 3.1. +3.3.1. IDENTIFYING A CRITICAL PAIR OF INPUTS +Set T = 1/(K∆2 +L) = β2L/(4K). +Let AK be any R-round collaborative algorithm. Let γ be a +transcript produced by AK. Let tr ≜ tr(γ) (r = 1, . . . , R) +be the time step at the end of the r-th round. We thus have +tR = T . For convenience, we define t0 = 1/K. Note that +t1, . . . , tR−1 are determined by γ, and t0 and tR are fixed +values. +We have the following simple fact. + +Tight Bounds for Collaborative Regret Minimization Multi-Armed Bandits +Fact 3.9. For any T > 0 and R > 0, and any transcript γ, +there is a r ∈ [R] such that +tr +tr−1 +≥ (KT ) +1 +R . +Define the following event Fr(γ) for r = 1, . . . , R. +Definition 3.10. Event Fr(γ): +For any i < r, it holds +that ti/ti−1 < (KT ) +1 +R ; and for i = r, we have tr/tr−1 ≥ +(KT ) +1 +R . +It is easy to see that {F1(γ), . . . , FR(γ)} are disjunctive +and they together partition the probability space. +For convenience of writing, let α ≜ log L +2λ . +For a transcript γ, let r = r(γ) be the round index such that +Fr(γ) holds. Let ℓ(γ) be the integer such that +β2(ℓ(γ)−1) +αK +≤ tr(γ)−1 < β2ℓ(γ) +αK . +(11) +The next claim shows that the value of ℓ(γ) will not be +larger than L. +Claim 3.11. For any γ, it holds that ℓ(γ) ≤ L. +Proof. By the definition of Fr(γ), we have +tr(γ)−1 +≤ +1 +K · (KT ) +r(γ)−1 +R +≤ +1 +K · +�β2L +4 +� R−1 +R +. +(12) +By (11) and (12), we have +β2(ℓ(γ)−1) +αK +≤ 1 +K · +�β2L +4 +�1− 1 +R +, +which implies +ℓ(γ) ≤ L − L +R + 1 + logβ α +2 +. +(13) +When L +R ≥ +logβ α +2 ++ 1 (recall the second inequality in (4)), +Inequality (13) implies ℓ(γ) ≤ L. +We now try to identify the index of the critical input pairs. +By the definition of Fr(γ), we have +tr(γ) ≥ (KT ) +1 +R · tr(γ)−1 = +�β2L +4 +� 1 +R +tr(γ)−1 . +(14) +Let mr(γ) = tr(γ) − tr(γ)−1 be the length of the r(γ)-th +round. By (11) and (14), we have +mr(γ) ≥ +��β2L +4 +� 1 +R +− 1 +� +β2(ℓ(γ)−1) +αK +. +(15) +Now consider a particular ℓ∗ = ℓ∗(AK) such that +Pr +Γ∼AK(I+ +L ,T ) +[ℓ(Γ) = ℓ∗] ≥ 1 +L . +(16) +Such an ℓ∗ must exist, since each transcript Γ = γ corre- +sponds to a unique r(γ) and consequently a unique ℓ(γ). +And by Claim 3.11, ℓ(γ) ≤ L always holds. +Our goal is show that the expected regret of AK is high on +either the input I+ +ℓ∗ or the input I− +ℓ∗. We call (I+ +ℓ∗, I− +ℓ∗) the +critical input pair for AK. +3.3.2. PROJECTION OF COLLABORATIVE ALGORITHM +Before analyzing the regret on the critical pair of inputs, we +first introduce a concept we call the projection of a collab- +orative algorithm on a single agent. +Let AK be a collaborative algorithm. For any k ∈ [K], we +use ProjAK +k +(I, ℓ) to denote a single-agent algorithm that +simulates AK as follows. Let γ be an arbitrary transcript +generated by running AK on I for T time steps. Let τ(γ, ℓ) +be the round index such that +β2(ℓ−1) +αK +≤ tτ(γ,ℓ)−1 < β2ℓ +αK . +(17) +And let +ζℓ = β2ℓ +α · β2( L +R −1) +8K +. +(18) +ProjAK +k +simulates AK as follows: In the first (τ(γ, ℓ) − 1) +rounds, at each time step t, if agents 1, . . . , K pull arms +a(1) +t , . . . , a(K) +t +in I respectively under AK, then ProjAK +k +pulls arms a(1) +t , . . . , a(K) +t +in I in order. In the τ(γ, ℓ)-th +round, at each time step t when t ≤ ζℓ, if agent k pulls arm +a(k) +t +in I under AK, then ProjAK +k +also pulls arm a(k) +t +in I. +For a transcript γ generated by running AK, we use +Projk(γ, ℓ) (k ∈ [K]) to denote the sequence of (jt, ot) +pairs in γ generated by the K agents in the round-robin +fashion in the first (τ(γ, ℓ) − 1) rounds, followed by the +first ζℓ of (jt, ot) pairs in the τ(γ, ℓ)-th round (or until the +end of the τ(γ, ℓ)-th round) in γ generated by agent k. +We further use Lastk(γ, ℓ) to denote the first ζℓ of (jt, ot) +pairs in the τ(γ, ℓ)-th round (or until the end of the τ(γ, ℓ)- +th round) in γ generated by the agent k. + +Tight Bounds for Collaborative Regret Minimization Multi-Armed Bandits +3.3.3. LARGE REGRET ON THE CRITICAL INPUT PAIR +Now we are ready to lower bound the regret. Let AK be +any K-agent collaborative algorithm, and ℓ∗ = ℓ∗(AK) +satisfying (16). +We define the following event for a transcript γ. +Definition 3.12. Event QAK(γ): ℓ(γ) = ℓ∗(AK). +We will abbreviate QAK(γ) to Q(γ) when AK is clear +from the context. By (16), we immediately have +Pr +Γ∼AK(I+ +L ,T ) +[Q(Γ)] ≥ 1 +L . +(19) +Let ΓL ∼ AK(I+ +L , T ). Let +Υ = {γ ∈ supp(ΓL) | Q(γ)}, +and for any k ∈ [K], +Υk(ℓ∗) = {Projk(γ, ℓ∗) | γ ∈ Υ}. +(20) +Note that for any γ ∈ Υ, by the definition of τ(γ, ℓ) +(Eq. (17)) and that of r(γ) (Eq. (11)), we have r(γ) = +τ(γ, ℓ(γ)). Therefore, by (18) and (15), we have +ζℓ∗ ≤ +��β2L +4 +� 1 +R +− 1 +� +β2(ℓ∗−1) +αK +≤ mτ(γ,ℓ∗). +(21) +Consequently, for any γ ∈ Υk(ℓ∗), +|γ| += +K · tτ(γ,ℓ∗)−1 + ζℓ∗ +≤ +K · β2ℓ∗ +αK + β2ℓ∗ +α +· β2( L +R −1) +8K +≤ +λβ2ℓ∗ +log L , +(22) +where the last inequality holds because β2( L +R −1) ≤ 8K +(recall the first inequality in (4)). +The following two claims exhibit properties of transcripts +in Υk(ℓ∗). The first claim states that the probability of a +random transcript being in Υk(ℓ∗) is significant. +Claim 3.13. For any I ∈ {I+ +ℓ∗, I− +ℓ∗} and any k ∈ [K], we +have +Pr +Γ∼Proj +AK +k +(I,ℓ∗) +[Γ ∈ Υk(ℓ∗) ∧ E(Γ)] ≥ e−2ǫ +2L . +Proof. By the definitions of projections of collaborative al- +gorithms and transcripts, we know that +Pr +Γ∼Proj +AK +k +(I+ +L ,ℓ∗) +[Γ ∈ Υk(ℓ∗)] = +Pr +Γ∼AK(I+ +L ,T ) +[Q(Γ)]. (23) +Note that both I∗ +L and I are in Iℓ. Using Lemma 3.8, setting +A = I+ +L , B = I, and q = |Γ| ≤ λβ2ℓ∗ +log L (by (22)), we have +Pr +Γ∼Proj +AK +k +(I+ +L ,ℓ∗) +[Γ ∈ Υk(ℓ∗)] +≤ e2ǫ +Pr +Γ∼Proj +AK +k +(I,ℓ∗) +[Γ ∈ Υk(ℓ∗)] +(24) +We thus have +Pr +Γ∼Proj +AK +k +(I,ℓ∗) +[Γ ∈ Υk(ℓ∗) ∧ E(Γ)] +Lemma 3.6 +≥ +Pr +Γ∼Proj +AK +k +(I,ℓ∗) +[Γ ∈ Υk(ℓ∗)] − 1 +L6 +(24) +≥ +e−2ǫ +Pr +Γ∼Proj +AK +k +(I+ +L ,ℓ∗) +[Γ ∈ Υk(ℓ∗)] − 1 +L6 +(23) += +e−2ǫ +Pr +Γ∼AK(I+ +L ,T ) +[Q(Γ)] − 1 +L6 +(19) +≥ +e−2ǫ +L +− 1 +L6 +≥ +e−2ǫ +2L . +The claim follows. +The next claim states that it is difficult to use a transcript in +Υk(ℓ∗) to differentiate inputs I+ +ℓ∗ (or I− +ℓ∗) from I+ +L . +Claim 3.14. For any I ∈ {I+ +ℓ∗, I− +ℓ∗} and any k ∈ [K], for +any γ ∈ Υk(ℓ∗) such that E(γ) holds, we have +Pr +Γ∼Proj +AK +k +(I,ℓ∗) +[Γ = γ] = cǫ +Pr +Γ∼AK(I+ +L ,T ) +[Projk(Γ, ℓ∗) = γ] +for some cǫ ∈ [e−2ǫ, e2ǫ]. +Proof. By the definitions of projections of collaborative al- +gorithms and transcripts, we have +Pr +Γ∼AK(I+ +L ,T ) +[Projk(Γ, ℓ∗) = γ] = +Pr +Γ∼Proj +AK +k +(I+ +L ,ℓ∗) +[Γ = γ] . +We thus only need to show that +Pr +Γ∼Proj +AK +k +(I,ℓ∗) +[Γ = γ] = cǫ +Pr +Γ∼Proj +AK +k +(I+ +L ,ℓ∗) +[Γ = γ] +for some cǫ ∈ [e−2ǫ, e2ǫ]. +It is easy to see that this +equality is a direct consequence of Lemma 3.8 (note that +|Γ| ≤ λβ2ℓ∗ +log L due to (22)). + +Tight Bounds for Collaborative Regret Minimization Multi-Armed Bandits +Let Γ+ +∼ +Ak(I+ +ℓ∗, T ), and Γ− +∼ Ak(I− +ℓ∗, T ). +By +Claim 3.14 and Claim 3.13, we know that +� +γ∈Υk(ℓ∗) +min +� +Pr[Projk(Γ+, ℓ∗) = γ], +Pr[Projk(Γ−, ℓ∗) = γ] +� +≥ e−8ǫ +2L . +(25) +For an input I and transcript γ = ((j1, o1), . . . , (j|γ|, o|γ|)), +let Reg(I, γ) denote the regret of pulling the arm sequence +j(γ) on the input I, that is, +Reg(I, γ) = +� +t=1,...,|γ| +(µ∗ − µjt). +For any transcript γ ∈ Υk(ℓ∗) and any k ∈ [K], we con- +sider the regret Uk = Reg +� +I+ +ℓ∗, Lastk(γ, ℓ∗) +� +and Vk = +Reg +� +I− +ℓ∗, Lastk(γ, ℓ∗) +� +. Due to our constructions of I+ +ℓ∗ +and I− +ℓ∗, we have for any k ∈ [K], +Uk + Vk ≥ ∆ℓ∗ · ζℓ∗ . +(26) +Since for k = 1, . . . , K, Lastk(γ, ℓ∗) are disjoint, we have +for any γ ∈ Υ, +Reg(I+ +ℓ∗, γ) + Reg(I− +ℓ∗, γ) +≥ +� +k∈[K] +(Uk + Vk) +≥ +K∆ℓ∗ · ζℓ∗ += +K∆ℓ∗ · β2ℓ∗ +α +· β2( L +R −1) +8K += +β2( L +R −1) +2α +· +1 +∆ℓ∗ . +(27) +By (25) and (27), we have that +max +� +E +� +Reg(AK(I+ +ℓ∗, T )) +� +, E +� +Reg(AK(I− +ℓ∗, T )) +�� +≥ +1 +2 · e−8ǫ +2L · β2( L +R −1) +2α +· +1 +∆ℓ∗ += +Ω +� +β +2L +R +L log L · +1 +∆ℓ∗ +� +. +This concludes the proof of Theorem 3.1. +4. The Algorithm +In this section, we design a batched algorithm for MAB, +which implies an algorithm for MAB in the CL model via +Observation 1.1. Our batched algorithm can be seen as a +variant of the one in (Gao et al., 2019). One notable differ- +ence is that our algorithm achieves an instance-dependent +Algorithm 1 BATCHEDMAB(I, λ, T ) +Initialize a set of active arms I0 ← I ; +set T0 ← 0 ; +for i = 1, 2, . . . , logλ T do +set Ti ← λi ; +set r ← 0 ; +while (r ≤ logλ T ) ∨ (|Ir| > 1) do +for a ∈ Ir do +make Tr − Tr−1 pulls on arm a ; +compute ˆµr +a, the estimated mean after Tr pulls ; +let ˆµr +max ← maxa∈Ir ˆµr +a ; +Ir+1 ← +� +a | ˆµr +max − ˆµr +a < 2 +� +ln(T 3|I|) +Tr +� +; +r ← r + 1 ; +if r < logλ T then +assign the rest of pulls to the single arm in Ir . +round complexity, while the round complexity in the algo- +rithm in (Gao et al., 2019) is worst case. +Our algorithm is described in Algorithm 4. It uses the suc- +cessive elimination method: It works in batches. In each +batch, we pull the remaining arms for an equal number of +times, and then eliminate those whose empirical means are +smaller than the best one by a good margin. +We have the following theorem. Due to space constraints, +we leave the proof of Theorem 4.1 to Appendix C. +Theorem 4.1. For any λ ≥ 2, BATCHEDMAB(I, λ, T ) +uses at most O +� +min +� +logλ +log(T N) +∆(I) , logλ T +�� +batches +with probability +� +1 − +1 +T 3 +� +and incurs an expected regret +of at most O +�� +a̸=⋆ +λ log(T N) +∆a +� +. +As mentioned, by Observation 1.1, the batched algorithm +immediately implies a collaborative algorithm. We thus +have the following corollary. +Corollary 4.2. There is a collaborative algorithm AK for +MAB such that under time horizon T , for any input I, +AK uses at most O +� +min +� +logλ +log(KT N) +∆(I) +, logλ(KT ) +�� +rounds with probability +� +1 − +1 +T 3 +� +and incurs an expected +regret of at most O +�� +a̸=⋆ +λ log(KT N) +∆a +� +. +Comparing with the Lower Bound. +We consider +the two-arm case in which ∆(I) += +∆a(a +̸= +⋆). +Same as that in the proof of Theorem 3.1, we con- +sider T += +1 +K(∆(I))2 , in which case logλ +log(KT ) +∆(I) += +Θ(logλ(KT )). Let R = logλ(KT ). We thus have λ = +(KT ) +1 +R . By Corollary 4.2, the expected regret is at most +O +� +(KT ) +1 +R log(KT ) · +1 +∆(I) +� +. +On the lower bound side, by Theorem 3.1, the expected + +Tight Bounds for Collaborative Regret Minimization Multi-Armed Bandits +regret is at least Ω +� +(KT ) +1 +R · +1 +log(KT ) log log(KT ) · +1 +∆(I) +� +. +Therefore, the upper and lower bounds match up to loga- +rithmic factors. +5. Concluding Remarks +In this paper, we give almost tight round-regrettradeoffs for +regret minimization in multi-armed bandits in the collabo- +rative learning model. Via a one-way connection between +the collaborative learning model and the batched learning +model, we also get an almost tight batch-regret tradeoff +for the same problem in the batched model; our results are +comparable with the state-of-the-art results in the batched +model. We believe that our lower bound analysis for MAB +in the CL model can be extended to N arms in a similar +way as it was done in the batched model (from two arms +(Perchet et al., 2015) to N arms (Gao et al., 2019)). The +analysis would be more involved though. +As far as we have concerned, our results are the first ones +for round-regret tradeoffs for regret minimization in ban- +dit problems in the CL model. It would be interesting to +study other regret minimization problems in bandits and re- +inforcement learning in the CL model. +References +Agarwal, A., Agarwal, S., Assadi, S., and Khanna, S. +Learning with limited rounds of adaptivity: Coin tossing, +multi-armed bandits, and ranking from pairwise compar- +isons. In COLT, pp. 39–75, 2017. +Bai, Y., Xie, T., Jiang, N., and Wang, Y.-X. Provably ef- +ficient q-learning with low switching cost. In NeurIPS, +2019. +Bistritz, I. and Leshem, A. Distributed multi-player bandits +- a game of thrones approach. In NeurIPS, pp. 7222– +7232, 2018. +Bubeck, S. and Budzinski, T. Coordination without com- +munication: optimal regret in two players multi-armed +bandits. +In Abernethy, J. D. and Agarwal, S. (eds.), +Conference on Learning Theory, COLT 2020, 9-12 July +2020, Virtual Event [Graz, Austria], volume 125 of Pro- +ceedings of Machine Learning Research, pp. 916–939. +PMLR, 2020. +Bubeck, S., Perchet, V., and Rigollet, P. Bounded regret +in stochastic multi-armed bandits. +In Shalev-Shwartz, +S. and Steinwart, I. (eds.), COLt 2013, volume 30 of +JMLR Workshop and Conference Proceedings, pp. 122– +134, 2013. +Esfandiari, H., Karbasi, A., Mehrabian, A., and Mirrokni, +V. S. Batched multi-armed bandits with optimal regret. +CoRR, abs/1910.04959, 2019. +Gao, Z., Han, Y., Ren, Z., and Zhou, Z. Batched multi- +armed bandits problem. In NeurIPS, 2019. +Hillel, E., Karnin, Z. S., Koren, T., Lempel, R., and +Somekh, O. Distributed exploration in multi-armed ban- +dits. In NIPS, pp. 854–862, 2013. +Jin, T., Shi, J., Xiao, X., and Chen, E. Efficient pure explo- +ration in adaptive round model. In NeurIPS, pp. 6605– +6614, 2019. +Jin, T., Tang, J., Xu, P., Huang, K., Xiao, X., and Gu, Q. Al- +most optimal anytime algorithm for batched multi-armed +bandits. In Meila, M. and Zhang, T. (eds.), Proceedings +of the 38th International Conference on Machine Learn- +ing, ICML 2021, 18-24 July 2021, Virtual Event, volume +139 of Proceedings of Machine Learning Research, pp. +5065–5073. PMLR, 2021. +Jun, K., Jamieson, K. G., Nowak, R. D., and Zhu, X. Top +arm identification in multi-armed bandits with batch arm +pulls. In AISTATS, pp. 139–148, 2016. +Kalkanli, C. and ¨Ozg¨ur, A. Batched thompson sampling. In +Ranzato, M., Beygelzimer, A., Dauphin, Y. N., Liang, P., +and Vaughan, J. W. (eds.), NeurIPS, pp. 29984–29994, +2021. +Karbasi, A., Mirrokni, V. S., and Shadravan, M. Paralleliz- +ing thompson sampling. In Ranzato, M., Beygelzimer, +A., Dauphin, Y. N., Liang, P., and Vaughan, J. W. (eds.), +NeurIPS, pp. 10535–10548, 2021. +Karpov, N. and Zhang, Q. Batched coarse ranking in multi- +armed bandits. In Larochelle, H., Ranzato, M., Hadsell, +R., Balcan, M., and Lin, H. (eds.), NeurIPS, 2020. +Karpov, N. and Zhang, Q. +Batched thompson sampling +for multi-armed bandits. CoRR, abs/2108.06812, 2021. +URL https://arxiv.org/abs/2108.06812. +Karpov, N. and Zhang, Q. +Communication-efficient +collaborative best arm identification, 2022a. +URL +https://arxiv.org/abs/2208.09029. +Karpov, N. and Zhang, Q. Collaborative best arm identifica- +tion with limited communication on non-iid data. CoRR, +abs/2207.08015, 2022b. +Karpov, N., Zhang, Q., and Zhou, Y. +Collaborative top +distribution identifications with limited interaction (ex- +tended abstract). In FOCS, pp. 160–171. IEEE, 2020. +Kittur, A., Chi, E. H., and Suh, B. Crowdsourcing user +studies with mechanical turk. In Czerwinski, M., Lund, +A. M., and Tan, D. S. (eds.), CHI, pp. 453–456. ACM, +2008. + +Tight Bounds for Collaborative Regret Minimization Multi-Armed Bandits +Landgren, P., Srivastava, V., and Leonard, N. E. Distributed +cooperative decision-making in multiarmed bandits: Fre- +quentist and bayesian algorithms. In CDC, pp. 167–172. +IEEE, 2016. +Landgren, P., Srivastava, V., and Leonard, N. E. Social +imitation in cooperative multiarmed bandits: Partition- +based algorithms with strictly local information. In CDC, +pp. 5239–5244. IEEE, 2018. +Liu, K. and Zhao, Q. Distributed learning in multi-armed +bandit with multiple players. IEEE Transactions on Sig- +nal Processing, 58(11):5667–5681, 2010. +Perchet, V., Rigollet, P., Chassang, S., and Snowberg, E. +Batched bandit problems. In COLT, pp. 1456, 2015. +R´eda, C., Vakili, S., and Kaufmann, E. Near-optimal col- +laborative learning in bandits. CoRR, abs/2206.00121, +2022. +doi: +10.48550/arXiv.2206.00121. +URL +https://doi.org/10.48550/arXiv.2206.00121. +Robbins, H. Some aspects of the sequential design of exper- +iments. Bulletin of the American Mathematical Society, +58(5):527–535, 1952. +Rosenski, J., Shamir, O., and Szlak, L. Multi-player ban- +dits - a musical chairs approach. In ICML, pp. 155–163, +2016. +Shi, C. and Shen, C. Federated multi-armed bandits. In +AAAI, pp. 9603–9611. AAAI Press, 2021. +Shi, C., Shen, C., and Yang, J. Federated multi-armed ban- +dits with personalization. In Banerjee, A. and Fukumizu, +K. (eds.), AISTATS, volume 130 of Proceedings of Ma- +chine Learning Research, pp. 2917–2925. PMLR, 2021. +Sz¨or´enyi, B., Busa-Fekete, R., Heged˝us, I., Orm´andi, R., +Jelasity, M., and K´egl, B. +Gossip-based distributed +stochastic bandit algorithms. In ICML, pp. 19–27, 2013. +Tao, C., Zhang, Q., and Zhou, Y. Collaborative learning +with limited interaction: Tight bounds for distributed ex- +ploration in multi-armed bandits. In Zuckerman, D. (ed.), +FOCS, pp. 126–146. IEEE Computer Society, 2019. +Thompson, W. R. On the likelihood that one unknown prob- +ability exceeds another in view of the evidence of two +samples. Biometrika, 25(3/4):285–294, 1933. +Wang, Y., Hu, J., Chen, X., and Wang, L. Distributed bandit +learning: Near-optimal regret with efficient communica- +tion. In ICLR. OpenReview.net, 2020. + +Tight Bounds for Collaborative Regret Minimization Multi-Armed Bandits +A. Mathematics Tools +Lemma A.1 (Hoeffding’s inequality). Let X1, . . . , Xn ∈ [0, 1] be independent random variables and X = 1 +n +�n +i=1 Xi. +Then +Pr[|X − E[X]| > t] ≤ 2 exp(−2t2n) . +Lemma A.2 (Azuma’s inequality). If the sequence of random variables Z0, . . . , Zn form a supermartingale and +∀t ∈ [n] : |Zt − Zt−1| ≤ d , +then +Pr[Zn − Z0 ≥ ǫ] ≤ exp +� −ǫ2 +2d2n +� +. +B. Missing Proofs +B.1. Proof of Lemma 3.3 +Proof. By (3), we can write +ln gA(γn) +gB(γn) = ln gA(γn−1) +gB(γn−1) + ln Pr[ΘA(jn) = on] +Pr[ΘB(jn) = on]. +We thus only need to show +����ln Pr[ΘA(jn) = on] +Pr[ΘB(jn) = on] +���� ≤ 5 +βℓ . +(28) +By the definition of Iσ +ℓ , both Pr[ΘA(jn) = on] and Pr[ΘB(jn) = on] are in the range +� +1 +2 − +1 +βℓ , 1 +2 + 1 +βℓ +� +. We thus have +ln 1 − 2β−ℓ +1 + 2β−ℓ ≤ ln Pr[ΘA(jn) = on] +Pr[ΘB(jn) = on] ≤ ln 1 + 2β−ℓ +1 − 2β−ℓ . +(29) +Using the fact that for any x ∈ +� +− 1 +2, 1 +2 +� +, x − x2 ≤ ln(1 + x) ≤ x, we have +ln 1 + 2β−ℓ +1 − 2β−ℓ += +ln(1 + 2β−ℓ) − ln(1 − 2β−ℓ) +≤ +2β−ℓ + 2β−ℓ + 4β−2ℓ +≤ +5β−ℓ. +(30) +Similarly, we have +ln 1 − 2β−ℓ +1 + 2β−ℓ += +ln(1 − 2β−ℓ) − ln(1 + 2β−ℓ) +≥ +−2β−ℓ − 4β−2ℓ − 2β−ℓ +≥ +−5β−ℓ. +(31) +Plugging (30) and (31) to (29), we get (28). +B.2. Proof of Lemma 3.4 +Proof. For any fixed transcript Γn−1 = γn−1 and algorithm A , Jn = jn is a deterministic value. +Using (B.1), we only need to prove for any jn ∈ {1, 2}, it holds that +E +On +� +ln Pr[ΘA(jn) = On] +Pr[ΘB(jn) = On] +� +≤ 11 +β2ℓ , +where On is the (random) reward of pulling the jn-th arm in I. + +Tight Bounds for Collaborative Regret Minimization Multi-Armed Bandits +Let δI, δA, δB be three values such that Pr[ΘI(jn) = 1] = 1/2 + δI, Pr[ΘA(jn) = 1] = 1/2 + δA, and Pr[ΘB(jn) = +1] = 1/2 + δB. By the property of inputs in I, we know that the absolute values of δI, δA, δB are at most 1/βℓ. +We immediately have Pr[ΘI(jn) = 0] = 1/2 − δI, Pr[ΘA(jn) = 0] = 1/2 − δA, and Pr[ΘB(jn) = 0] = 1/2 − δB. We +also have Pr[On = 1] = 1/2 + δI, and Pr[On = 0] = 1/2 − δI. +With the above notations, we write +E +On +� +ln Pr[ΘA(jn) = On] +Pr[ΘB(jn) = On] +� += +�1 +2 + δI +� +ln 1 + 2δA +1 + 2δB ++ +�1 +2 − δI +� +ln 1 − 2δA +1 − 2δB += +1 +2 ln 1 − 4δ2 +A +1 − 4δ2 +B ++ δI ln (1 + 2δA)(1 − 2δB) +(1 + 2δB)(1 − 2δA). +(32) +We bound the two terms in (32) separately. For the first term, +ln 1 − 4δ2 +A +1 − 4δ2 +B += +ln(1 − 4δ2 +A) − ln(1 − 4δ2 +B) +≤ +−4δ2 +A + 4δ2 +B + 16δ4 +B. +(33) +For the second term, +ln (1 + 2δA)(1 − 2δB) +(1 + 2δB)(1 − 2δA) +≤ +2δA − (−2δA − 4δ2 +A) − 2δB − (2δB − 4δ2 +B) += +4δA + 4δ2 +A − 4δB + 4δ2 +B. +(34) +Plugging (33) and (34) to (32), we have +E +On +� +ln Pr[ΘA(jn) = On] +Pr[ΘB(jn) = On] +� +≤ +(−2δ2 +A + 2δ2 +B + 8δ4 +B) + δI(4δA + 4δ2 +A − 4δB + 4δ2 +B) +≤ +11 +β2ℓ , +where the last inequality is due to |δA| , |δB| , |δI| ≤ +1 +βℓ . +C. Proof of Theorem 4.1 +Correctness. +Let N = |I| be the number of arms in the input I. We define the following event: +E2 : +∀a ∈ I, r ≤ logλ T, |ˆµr +a − µa| ≤ +� +ln(T 3N) +Tr +. +The following lemma states that +Lemma C.1. Pr[E2] ≥ 1 − 1/T 3. +Proof. By Hoeffding’s inequality (Lemma A.1), we have for any a ∈ I, for any r ≤ logλ T , it holds that +Pr + +|ˆµr +a − µa| > +� +ln(T 3N) +Tr + + +≤ +2 exp +�−2 ln(T 3N) +Tr +· Tr +� +≤ +2 +T 6N 2 . + +Tight Bounds for Collaborative Regret Minimization Multi-Armed Bandits +By a union bound, we have +Pr[E2] ≥ 1 − N · (1 + logλ T ) · +2 +T 6N 2 ≥ 1 − 1 +T 3 . +Lemma C.2. If E2 holds, then for any r we have ⋆ ∈ Ir. +Proof. By the definition of event E2, we have for any b ∈ I, +ˆµr +b − ˆµr +⋆ +< +(µb − µ⋆) + 2 +� +ln(T 3N) +Tr +≤ 2 +� +ln(T 3N) +Tr +. +Therefore, ⋆ ∈ Ir based on the description of Algorithm 4. +For a suboptimal arm a, define r(a) to be the smallest value such that Tr > 64 ln(T 3N) +∆2a +. The next lemma shows that all +suboptimal arms will be eliminated by before the r(a)-th round ends. +Lemma C.3. If E2 holds, then any arm a ̸= ⋆ does not appear in any Ir for r > r(a). +Proof. For any fixed arm a ̸= ⋆, we consider the case that a ∈ Ir(a), in which case the arm a will be pulled for Tr(a) times. +Abbreviating r(a) as r, we have +ˆµr +max − ˆµr +a − 2 +� +ln(T 3N) +Tr +≥ +ˆµr +⋆ − ˆµr +a − 2 +� +ln(T 3N) +Tr +(definition of ˆµr +max). +≥ +µ⋆ − µa − 4 +� +ln(T 3N) +Tr +(E2 holds) += +∆a − 4 +� +ln(T 3N) +Tr +≥ +∆a − 4 · ∆a +8 +(definition of Tr) += +∆a +2 +> 0 , +which means that arm a will be eliminated in the r(a)-th round. +By Lemma C.3 and the fact that Tr+1 ≤ λTr for any r, the number of pulls on each suboptimal arm is bounded by +Tr(a) ≤ 64λ ln(T 3N) +∆2a +≤ 200λ ln(T N) +∆2a +. +(35) +Hence, the expected regret is bounded by +� +a̸=⋆ +Tr(a)∆a ≤ +� +a̸=⋆ +200λ ln(T N) +∆a +. +(36) + +Tight Bounds for Collaborative Regret Minimization Multi-Armed Bandits +Round Complexity. +We next bound the number of batches. After the r-th round, the algorithm makes at least λr pulls. +Therefore, the number of rounds is upper bounded by logλ T with certainty. +On the other hand, if there is some round r for which |Ir| = 1, then we will pull this arm until the end of the time horizon. +By Lemma C.3, all suboptimal arms will be pruned after maxa̸=⋆ r(a) rounds. Therefore, if E2 holds, then the number of +rounds can also be bounded by +logλ +64 ln(T 3N) +(∆(I))2 ++ 1 = O +� +logλ +ln(T N) +∆(I) +� +. +Therefore, the total number of rounds is bounded by O +� +min +� +logλ +ln(T N) +∆(I) , logλ T +�� +with probability +� +1 − +1 +T 3 +� +. + diff --git a/H9FJT4oBgHgl3EQfFSyX/content/tmp_files/load_file.txt b/H9FJT4oBgHgl3EQfFSyX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e646bba395355928c645185787fbc7647d9ce81c --- /dev/null +++ b/H9FJT4oBgHgl3EQfFSyX/content/tmp_files/load_file.txt @@ -0,0 +1,878 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf,len=877 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content='11442v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content='LG] 26 Jan 2023 Collaborative Regret Minimization in Multi-Armed Bandits Nikolai Karpov 1 Qin Zhang 1 Abstract In this paper, we study the collaborative learning model, which concerns the tradeoff between par- allelism and communication overhead in multi- agent reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=' For a fundamen- tal problem in bandit theory, regret minimiza- tion in multi-armed bandits, we present the first and almost tight tradeoffs between the number of rounds of communication between the agents and the regret of the collaborative learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=' Introduction One of the biggest challenges with reinforcement learn- ing is scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=' In recent years, a series of papers (Tao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=' Karpov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=' Karpov & Zhang, 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content='b) studied bandit problems in the collaborative learning (CL) model, where multiple agents interact with the environment to learn simultaneously and cooperatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=' One of the most expensive resources in the CL model is communication, which consists of the number of communication steps (round complexity) and the total bits of messages exchanged between agents (bit complex- ity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=' Communication directly contributes to learning time (network bandwidth constraints and latency), energy con- sumption (communication is frequently the biggest energy drain for tasks such as deep-sea/outer-space exploration), as well as data usage (if messages are sent by mobile de- vices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=' In this paper, we focus on the round complexity in the CL model and consider a basic problem in the bandit theory named regret minimization in multi-armed bandits (MAB for short).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=' We give almost tight round-regret trade- offs for MAB in the CL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=' The CL model is closely related to the batched learning model, which has recently received considerable attention in bandit theory and reinforcement learning (Perchet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=' Jun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=' Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=' Esfandiari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=' Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=' Karpov & Zhang, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=' The batched model is motivated by applications in which there is a sig- 1Computer Science Department, Indiana University, Bloom- ington, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FJT4oBgHgl3EQfFSyX/content/2301.11442v1.pdf'} +page_content=' Correspondence to: Nikolai Karpov